Introduction↩︎

High degree hypersurfaces and complete intersections in projective space are, by most measures, geometrically complicated. For instance, unlike the simplest of algebraic varieties [1], high degree hypersurfaces over the complex numbers contain very few—if any!—rational subvarieties: see [2]–[5]. New phenomena appear, however, in positive characteristic \(p > 0\), complicating the correlation between degree and geometric complexity. The Fermat hypersurface \[X \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{x_0^{q+1} + \cdots + x_n^{q+1} = 0\} \subset \mathbf{P}^n\] of degree \(q + 1\), with \(q = p^e\) an integer power, is an exemplar: Many properties of \(X\) are reminiscent of those typically expected in quadric hypersurfaces: it is projectively self-dual and its Gauss map is a homeomorphism [6], [7]; its smooth hyperplane sections have constant moduli [8]; it has many linear spaces [9]; and, if \(n \geq 3\), it is even unirational [10]! Traditionally, these curiosities are explained ad hoc by identifying the equation of \(X\) with a Hermitian form for the quadratic extension \(\mathbf{F}_{q^2} \mid \mathbf{F}_q\), as in [11]–[13].

In contrast, I view the quadratic nature of \(X\) as the confluence between the almost linear nature of \(q\)-powers in characteristic \(p\) and the special form of the equation of \(X\). This perspective places this Fermat hypersurface, a historically isolated example, in a broad context and suggests a systematic way of identifying related phenomena. The approach taken here is as follows:

A profile with respect to \(q\) is an integer polynomial \(a(t) = a_0 + a_1 t + \cdots + a_m t^m \in \mathbf{Z}_{\geq 0}[t]\) subject to a technical condition, given in [qatics-profiles], which is satisfied, for example, if \(a_j \leq q - 1\) for each \(j = 0,\ldots,m\). A polynomial with coefficients a field \(\mathbf{k}\) of characteristic \(p\) is said to have profile \(a(t)\) with respect to \(q\)—or simply is called a \((q;a)\)-tic polynomial—if it is of the form \[f(x_0,\ldots,x_n) = \sum\nolimits_{i \in I} \prod\nolimits_{j = 0}^m f_{i,j}(x_0,\ldots,x_n)^{q^j} \in \mathbf{k}[x_0,\ldots,x_n]\] where each \(f_{i,j}\) is a homogeneous polynomial of degree \(a_j\). The form of this expression is invariant under linear changes of variables, so \((q;a)\)-tic polynomials span a canonical linear subspace within the space of polynomials of degree \(a(q) = a_0 + a_1 q + \cdots + a_m q^m\). The vanishing locus in \(\mathbf{P}^n\) of a \((q;a)\)-tic polynomial is a \((q;a)\)-tic hypersurface. The Fermat hypersurface above is an example with profile \(a(t) = 1 + t\) with respect to \(q\). The purpose of this works is to develop the following principle: \((q;a)\)-tic hypersurfaces behave as if they were hypersurfaces of degree \(a(1 + \varepsilon) \approx a_0 + a_1 + \cdots + a_m\) for some small real number \(\varepsilon > 0\).

One aspect of this principle is that hypersurfaces sharing the same profile should be treated as a single family. Concretely, one looks for properties of \((q;a)\)-tic hypersurfaces which may be expressed in terms of the profile \(a\), ideally independently of, or at least uniformly in, the prime power \(q\). A simple illustration of this is found in the geometry of linear spaces:

This is a summary of 1, 1, and 1 in the case of a hypersurface, all of which are formulated and proven without the restriction \(a \neq 2t^m\) and more generally for \((q;\mathbf{a})\)-tic complete intersections, vanishing loci of a regular sequence of \((q;a)\)-tic polynomials with \(a\) ranging over a multi-set of profiles \(\mathbf{a}\). These generalize classical results on Fano schemes of complete intersections and, once the techniques developed for handling \((q;a)\)-tic equations are developed in §§1–2, their proofs essentially follow the strategy from the classical case found in [14] and [15].

A second aspect of this principle is, of course, that \((q;a)\)-tic hypersurfaces whose profile is small compared to its dimension should be geometrically simple, regardless of the prime power \(q\). Theorem 1 is some evidence in this direction since the dimension of the Fano scheme depends only on \(a\) and not on \(q\). To explain a second result in this direction, recall classical results of Morin and Predonzan from [16], [17] which show that a general complete intersection in \(\mathbf{P}^n\) is unirational whenever its total degree \(d\) is much smaller than \(n\); see [18] for a classical source, but also [19] for a succinct exposition in modern language and [20] for an improved bound. An analogue of this for \((q;\mathbf{a})\)-tic complete intersections is:

This is proven in §7 via an inductive argument, using the constructions developed in §§4–6. A similar result, formulated and proven in a different language, seems to appear in [21]; see also the related result in [22]. The integer \(n_0(\mathbf{a})\) can be computed for small \(\mathbf{a}\): see [unirationality-estimates]. The result is already interesting when \(\mathbf{a} = (d)\) consists of a single constant polynomial, this is a statement about unirationality of hypersurfaces of degree \(d\), in which case the integer \(n_0(d)\) appearing here is on the order of \(2^{d2^d}\), significantly improving past constructions: see the companion paper [23].

Briefly, the unirationality construction of Morin and Predonzan goes as follows: given a general complete intersection \(X \subset \mathbf{P}^n\) of multi-degree \(\mathbf{d} = (d_1,\ldots,d_c)\), projection away from a general \(r\)-plane in \(X\) yields a fibration \(\widetilde{X} \to \mathbf{P}^{n-r-1}\) whose generic fibre \(X'\) is itself a complete intersection of multi-degree \(\mathbf{d}' = (d_1 - 1, \ldots, d_c - 1)\) in a \(\mathbf{P}^{r+1}\). With an appropriate choice of \(r\) and \(n\) depending on \(\mathbf{d}\), it is possible to find a base change of \(X'\) that carries a large linear space, allowing the argument to proceed inductively. This strategy does not adapt in a straightforward manner for \((q;\mathbf{a})\)-tic complete intersections because the generic fibre of the projection \(\widetilde{X} \to \mathbf{P}^{n-r-1}\) does not seem to have structure that is captured by the theory of profiles: see [families-not-strict-transform] for an explicit example.

Instead, Theorem 2 is established with a generalization of an old unirationality construction for cubic hypersurfaces found in [24] and [25] which is based on the following observation: Given a hypersurface \(X \subset \mathbf{P}^n\) of degree \(d\), the space of penultimate tangents \[X' = \{ (x,[\ell]) : \ell \subset \mathbf{P}^n\;\text{a line intersecting \(X\) at \(x\) with multiplicity \(\geq d-1\)} \}\] is generically a family of complete intersections of multi-degree \(\mathbf{d}' = (d-2,d-3,\ldots,1)\) over \(X\). Moreover, for \(X\) general, there is a dominant rational map \(X' \dashrightarrow X\) sending \((x,[\ell])\) to the residual point of intersection \(x' = X \cap \ell - (d-1)x\). By restricting \(X'\) to a sufficiently large and general linear space in \(X\), unirationality can be established by induction on a poset consisting of all multi-degrees. In particular, it is essential to prove the result for all complete intersections.

Much effort is made to perform the constructions in §§4–7 globally, so that they work well in families, albeit introducing additional technicalities. The hope is to eventually make the generality conditions on \(X\) in Theorems 1 and 2 effective \(n\) sufficiently large. In characteristic \(0\), the works [26], [27] show that the Fano schemes \(\mathbf{F}_r(X)\) of every smooth hypersurface \(X \subset \mathbf{P}^n\) of degree \(d\) is irreducible of its expected dimension once \(n \geq 2\binom{d+r-1}{r} + r\). This may then be used in a refinement of the Morin construction to show that every smooth hypersurface of degree \(d\) is unirational once \(n \geq 2^{d!}\). While it is possible that these results may be extended to characteristics \(p > d\), Theorem 1 shows that these statements cannot hold verbatim when \(p \leq d\) wherein there exists a nontrivial profile \(a\) with \(a(p) = d\): see [qatics-ordering-properties][qatics-ordering-properties46degrees]. An optimistic guess would be that the appearance of these canonical sub-linear systems is the only issue which arises, suggesting the following:

Note that \(n_0(a,r)\) does not depend on \(p\), so that Conjecture 3 predicts a uniform phenomenon for varying characteristic. Unfortunately, this statement cannot be quite right for higher prime powers \(q\), since there will be a profile with respect to \(p\) with larger expected dimension. A result in this direction would be a first step toward an effective statement for unirationality.

Further questions. — This work is primarily focused on the linear projective geometry of \((q;a)\)-tic hypersurfaces, and so many basic questions and properties remain to be explored: automorphisms, moduli, singularities, and so forth. Three main directions that seem the most interesting are:

First, the principle that \((q;a)\)-tic hypersurfaces behave as if they were of degree \(a(1 + \varepsilon)\) is taken to be a rough qualitative heuristic, and it would be interesting to make this more precise and quantitative. For instance, the work in [9], [28] shows that the geometry of lines in hypersurfaces of profile \(a(t) = 1 + t\), or \(q\)-bic hypersurfaces, is reminiscent of that of cubics.

Second, the theory developed here is extrinsic in that \((q;a)\)-tic-ness is a structure with which an object may be equipped with. Might there be an intrinsic characterization? One instance occurs with \(q\)-bic hypersurfaces, wherein [29] show that they are characterized amongst hypersurfaces of the same degree as those having the smallest \(F\)-pure threshold. From a different direction, many examples suggest that non-\(F\)-splitness is related to the special form of the defining equations: see [30]–[33] for a few examples.

Third, the definition in §2 of \((q;a)\)-tic polynomials generalize to provide canonical linear systems in \(\Gamma(Y,\mathcal{L}^{\otimes a(q)})\) for any scheme \(Y\) and \(\mathcal{L} \in \mathop{\mathrm{Pic}}Y\). The situation here is the case \((Y,\mathcal{L}) = (\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n}(1))\), and other cases that have occurred in the literature include Deligne–Lusztig varieties [34] and the Frobenius incidence correspondence of [35]. It would be interesting to study the special properties of the divisors and linear systems constructed in this fashion.

Outline. — §1 develops the combinatorics of profiles and §2 gives a definition of \((q;a)\)-tic hypersurfaces and studies their basic properties. Fano schemes of \((q;\mathbf{a})\)-tic complete intersections are studied in §3. Families of \((q;\mathbf{a})\)-tic complete intersections are defined in §4. The penultimate tangent construction for families is made in §5 and the corresponding residual point map is studied in §6. Finally, the unirationality result is proven in §7.

Notation. — Throughout, \(\mathbf{k}\) denotes an algebraically closed field of characteristic \(p > 0\) and \(q \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =p^e\) is a positive integer power of \(p\). Unless otherwise stated, \(V\) denotes a \(\mathbf{k}\)-vector space of dimension \(n+1\). Write \(\mathop{\mathrm{Fr}}\colon \mathbf{k}\to \mathbf{k}\) for the \(q\)-power Frobenius morphism and, for any \(\mathbf{k}\)-vector space \(W\), let \(W^{[1]} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{k}\otimes_{\mathop{\mathrm{Fr}}, \mathbf{k}} W\) be its Frobenius twist. Set \(W^{[0]} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =W\) and, for each integer \(i \geq 1\), inductively define \(W^{[i+1]} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =(W^{[i]})^{[1]}\). Schemes are all taken to be over \(\mathbf{k}\) and \(\mathbf{P}V\) is the projective space of lines in \(V\).

Acknowledgements. — Thanks to Jan Lange, Matthias Schütt, and Noah Olander for their interest and helpful conversations related to this work. I was supported by a Humboldt Postdoctoral Research Fellowship during the completion of this work.

1 Profiles↩︎

A profile is a discrete invariant associated with a polynomial in positive characteristic \(p > 0\) which records its special shape with respect to a positive integer power \(q = p^e\) of the characteristic. Profiles are subject to a technical condition which ensures a certain uniqueness in writing polynomials in their distinguished form: see 1. While this hypothesis is not strictly necessary at this point, it often becomes important in applications. This section defines profiles and studies their combinatorics, especially various order relations amongst profiles.

1.1 Profiles↩︎

— Given an integer polynomial \(a(t) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =a_0 + a_1 t + \cdots + a_m t^m\) with nonnegative coefficients, consider the tensor functor defined by \[\mathrm{S}^a(V^\vee) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \bigotimes\nolimits^m_{j = 0} \mathop{\mathrm{Sym}}^{a_j}(V^\vee)^{[j]}.\] Identifying \(\mathop{\mathrm{Sym}}^{a_j}(V^\vee)^{[j]}\) with the \(\mathbf{k}\)-linear subspace of \(\mathop{\mathrm{Sym}}^{a_j q^j}(V^\vee)\) consisting of \(q^j\)-powers provides a multiplication map \[\operatorname{mult} \colon \mathrm{S}^a(V^\vee) \to \mathop{\mathrm{Sym}}^{a(q)}(V^\vee).\] Call \(a(t)\) a profile with respect to the prime power \(q\) if this multiplication map is injective for all finite-dimensional \(\mathbf{k}\)-vector spaces \(V\). Denote by \[\mathbf{Prfl}\mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \{ a \in \mathbf{Z}_{\geq 0}[t] : \mathop{\mathrm{mult}}\colon \mathrm{S}^a(V^\vee) \to \mathop{\mathrm{Sym}}^{a(q)}(V^\vee) \;\text{is injective for all finite-dimensional \(V\)} \}\] the set of profiles with respect to \(q\). Given \(a \in \mathbf{Prfl}\) and a vector space \(V\), \(\mathrm{S}^a(V^\vee)\) will often tacitly be identified with its image in \(\mathop{\mathrm{Sym}}^{a(q)}(V^\vee)\) under the injective multiplication map.

Whether or not a given element of \(\mathbf{Z}_{\geq 0}[t]\) is a profile depends on \(q\): For instance, \(a(t) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =t + q\) is not a profile with respect to \(q\) since, for a \(2\)-dimensional vector space \(V^\vee = \mathbf{k}\cdot u \oplus \mathbf{k}\cdot v\), the elements \(u^{[1]} \otimes v^q\) and \(v^{[1]} \otimes u^q\) are distinct in \(\mathrm{S}^a(V^\vee)\), but have the same image in \(\mathop{\mathrm{Sym}}^{2q}(V^\vee)\). The same polynomial \(t + q\) is, however, a profile with respect to \(q^k\) for any \(k \geq 2\). In what follows, however, \(q\) remains fixed, the dependence of \(\mathbf{Prfl}\) on \(q\) will be suppressed.

1.2 Partial orderings on profiles↩︎

— Endow \(\mathbf{Prfl}\) with two partial orderings as follows: First, viewing a profile as a sequence of nonnegative integers provides the product ordering \(\preceq\), where for profiles \(a \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\sum\nolimits_{j \geq 0} a_j t^j\) and \(b \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\sum\nolimits_{j \geq 0} b_j t^j\), \[a \preceq b \;\;\iff\;\; a_j \leq b_j\;\;\text{for all \(j \geq 0\)}.\] Second, identifying a \(a \in \mathbf{Prfl}\) with the space \(\mathrm{S}^a(V^\vee)\) provides the containment ordering \(\sqsubseteq\), where \(a \sqsubseteq b\) if and only if \(a(q) = b(q) = \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}}d\) and \[\mathrm{S}^a(V^\vee) \subseteq \mathrm{S}^b(V^\vee) \subseteq \mathop{\mathrm{Sym}}^d(V^\vee) \; \text{for all finite-dimensional \(\mathbf{k}\)-vector spaces \(V\),}\] where the tensor functors are identified with their image under the multiplication map.

1.3 Examples and properties↩︎

— Regarding the partial orderings \(\preceq\) and \(\sqsubseteq\) on \(\mathbf{Prfl}\):

1. The product ordering is strictly monotonic for numerical degrees: If \(a \prec b\), then \(a(q) < b(q)\). The containment ordering, on the other hand, splits \(\mathbf{Prfl}\) into connected components \[\mathbf{Prfl}_d \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{a \in \mathbf{Prfl}: a(q) = d\}\] indexed by numerical degrees. Each \(\mathbf{Prfl}_d\) has a unique \(\sqsubseteq\)-maximal element given by the constant profile \(a_{d\text{-}\mathrm{max}}(t) = d\), and a unique \(\sqsubseteq\)-minimal element given by \(a_{d\text{-}\mathrm{min}}(t) = a_0 + a_1t + \cdots + a_mt^m\) with coefficients \(0 \leq a_j \leq q - 1\) arising from the base \(q\) expansion of \(d\).

2. More interestingly, both orderings are strictly monotonous for coefficient sums. This is clear for \(\preceq\). To see that \(a \sqsubset b\) implies \(a(1) < b(1)\), observe that \(\mathrm{S}^a(V^\vee) \subset \mathrm{S}^b(V^\vee)\) for varying vector spaces \(V\) gives the dimensional inequality \[\prod\nolimits_{j \geq 0} \binom{a_j + n}{a_j} < \prod\nolimits_{j \geq 0} \binom{b_j + n}{b_j} \;\;\text{for all \(n \geq 0\)}.\] Using that \(\binom{n}{k} \sim \frac{n^k}{k!}\) for large \(n\) and fixed \(k\), taking logarithms, and doing away with constants shows that for all \(n \gg 0\), \[\sum\nolimits_{j \geq 0} a_j \cdot \log(a_j + n) < \sum\nolimits_{j \geq 0} b_j \cdot \log(b_j + n).\] By taking \(n\) even larger, \(\log(n)\) may be made arbitrarily close to \(\log(c + n)\) for any constant \(c\). Since there are only finitely many constants \(a_j\) and \(b_j\), this implies that \(a(1) < b(1)\).

3. Neither \(\preceq\) nor \(\sqsubseteq\) give total orderings on \(\mathbf{Prfl}\). This remains the case for \(\sqsubseteq\) even upon restriction to a connected component \(\mathbf{Prfl}_d\): for instance, \[a(t) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =t^2 + (q+1) \;\;\text{and}\;\; b(t) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =(q+1)t + 1\] are profiles of numerical degree \(d = q^2+q+1\) which are \(\sqsubseteq\)-incomparable.

4. Call \(b \in \mathbf{Prfl}\) nonreduced if its constant term is zero: \(b(0) = 0\). Any profile \(a\) preceding a nonreduced profile \(b\) in either ordering is also nonreduced.

5. Both orderings are compatible with addition in the following sense: For \(a, b, c \in \mathbf{Prfl}\) such that \(a + c\) and \(b + c\) are also profiles, \[\text{if}\; a \sqsubseteq b,\; \text{then}\; a + c \sqsubseteq b + c,\] and similarly for \(\preceq\). This is straightforward to see for \(\preceq\). For \(\sqsubseteq\), let \(V\) be a \(\mathbf{k}\)-vector space and let \(f \in \mathrm{S}^{a+c}(V^\vee)\). Then there exists an expansion of the form \[f = \sum\nolimits_{i \in I} g_i h_i \;\;\text{with}\; g_i \in \mathrm{S}^a(V^\vee) \;\text{and}\; h_i \in \mathrm{S}^c(V^\vee).\] Viewing the \(g_i\) as elements of \(\mathrm{S}^b(V^\vee)\) shows that \(f \in \mathrm{S}^{b+c}(V^\vee)\).

A third ordering \(\rightsquigarrow\) on \(\mathbf{Prfl}\) obtained by combining the two will be also be useful: For \(a, b \in \mathbf{Prfl}\), \[a \rightsquigarrow b \iff \text{there exists \(a' \in \mathbf{Z}_{\geq 0}[t]\) such that \(a \preceq a' \sqsubseteq b\)}.\] Here, the relations \(\preceq\) and \(\sqsubseteq\) are extended to \(\mathbf{Z}_{\geq 0}[t]\) in the natural way: \(a \preceq a'\) means that each coefficient of \(a'\) is at least that of \(a\), and \(a' \sqsubseteq b\) means that the image of \(\mathrm{S}^{a'}(V^\vee)\) under the multiplication map is contained in \(\mathrm{S}^b(V^\vee)\) for all finite-dimensional \(\mathbf{k}\)-vector spaces \(V\).

These partial orderings, \(\preceq\) in particular, make it simple to formulate a criterion for when a nonnegative integer polynomial \(a \in \mathbf{Z}_{\geq 0}[t]\) is a profile with respect to \(q\):

A simple but very useful consequence of this characterization used tacitly throughout—in 1 for instance—is the following

2 \((q;\mathbf{a})\)-tic schemes↩︎

Each \(a \in \mathbf{Prfl}\) determines a canonical linear system \[\Gamma(Y,\mathcal{L}^{\otimes a}) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \mathop{\mathrm{image}}( \mathop{\mathrm{mult}}\colon \mathrm{S}^a\Gamma(Y,\mathcal{L}) \to \Gamma(Y,\mathcal{L}^{\otimes a(q)}))\] for every \(\mathbf{k}\)-scheme \(Y\) and \(\mathcal{L} \in \mathop{\mathrm{Pic}}Y\); note that for general \((Y,\mathcal{L})\), the displayed multiplication map may not be injective. Divisors in the corresponding linear series tend to be special compared to the general divisor in the complete linear series associated with \(\mathcal{L}^{\otimes a(q)}\) due to the special form of its equations. One way to access these special properties is by carrying as additional structure a lift of its defining section to the tensor product space \(\mathrm{S}^a\Gamma(Y,\mathcal{L})\). This article is concerned primarily with the case of projective \(n\)-space \(Y = \mathbf{P}V\) and \(\mathcal{L} = \mathcal{O}_{\mathbf{P}V}(1)\), and this section describes some of the basic properties of complete intersections of such hypersurfaces.

2.1 Definitions↩︎

— Let \(a \in \mathbf{Prfl}\) be a profile.

• A \((q;a)\)-tic tensor is an element \(\alpha \in \mathrm{S}^a(V^\vee)\).

• A \((q;a)\)-tic polynomial is the image \(f_\alpha \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathop{\mathrm{mult}}(\alpha)\) of a \((q;a)\)-tic tensor under multiplication.

• A \((q;a)\)-tic hypersurface is the zero locus \(X_\alpha \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathrm{V}(f_\alpha)\) in \(\mathbf{P}V\) of a nonzero \((q;a)\)-tic polynomial.

More generally, given a multi-profile \(\mathbf{a}\)—a finite multi-set consisting of elements of \(\mathbf{Prfl}\)—let \[\mathrm{S}^{\mathbf{a}}(V^\vee) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \bigoplus\nolimits_{a \in \mathbf{a}} \mathrm{S}^a(V^\vee)\] be the space of \((q;\mathbf{a})\)-tic tensors. Given such a tensor \(\boldsymbol{\alpha}\), the vanishing locus \(X_{\boldsymbol{\alpha}} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathrm{V}(f_\alpha : \alpha \in \boldsymbol{\alpha}) \subseteq \mathbf{P}V\) of its associated polynomials is called a \((q;\mathbf{a})\)-tic scheme; when the sequence of defining polynomials form a regular sequence, \(X_{\boldsymbol{\alpha}}\) is called a \((q;\mathbf{a})\)-tic complete intersection.

2.2 Examples↩︎

— Given a profile \(a = a_0 + a_1 t + \cdots + a_m t^m\), a \((q;a)\)-tic polynomial is of the form \[f = \sum\nolimits_{i \in I} g_{i,0} \cdot (g_{i,1})^q \cdots (g_{i,m})^{q^m}\] where \(g_{i,j}\) is a homogeneous polynomial of degree \(a_j\). Some simple instances include:

1. The constant polynomial \(a(t) = d\) is a profile for any prime power \(q\), and \((q;a)\)-tic polynomial is synonymous for degree \(d\) polynomial. In particular, if \(a(t) = 1\) is a linear profile, then \((q;a)\)-tic polynomials are linear polynomials.

2. If a profile \(a(t) = a_1t + \cdots + a_mt^m\) has zero constant term, so it is nonreduced, then any \((q;a)\)-tic polynomial is a \(\mathbf{k}\)-linear combination of \(q\)-power monomials, and its associated hypersurface is geometrically nonreduced.

3. \(a(t) = 1 + t + \cdots + t^m\) is a profile for any \(q\), and \((q;a)\)-tic polynomials are of the form \[f(x_0,\ldots,x_n) = \sum\nolimits_{i_0,\ldots,i_m = 0}^n c_{i_0,i_1,\ldots,i_m} \cdot \prod\nolimits_{j = 0}^m (x_{i_j})^{q^j} \;\; \text{for scalars \(c_{i_0,i_1,\ldots,i_m} \in \mathbf{k}\)}.\] In particular, when \(a(t) = 1 + t\), these are the \(q\)-bic hypersurfaces from [9, p. 1.6], and the underlying \((q;a)\)-tic tensor is but a \(q\)-bic form in the sense of [36, p. 1.1].

4. \(a(t) = 2 + t\) is a profile whenever \(q \neq 2\), and \((q;a)\)-tic polynomials are of the form \[f(x_0,\ldots,x_n) = \sum\nolimits_{0 \leq i \leq j \leq n} \sum\nolimits_{k = 0}^n c_{ijk} \cdot x_i x_j x_k^q \;\; \text{for scalars \(c_{ijk} \in \mathbf{k}\)}.\]

5. The Fermat polynomial \(x_0^d + \cdots + x_n^d\) of degree \(d\) is a \((q;a)\)-tic polynomial for any profile \(a\) of numerical degree \(a(q) = d\). This provides a cheap proof of the fact that the general \((q;a)\)-tic hypersurface is smooth whenever its numerical degree is coprime to the characteristic \(p\); see 1 for another argument in the general case.

A simple but extremely useful observation is that a \((q;\mathbf{a})\)-tic structure is preserved upon passing to linear sections. The following statement follows easily from functoriality of tensor functors:

2.3 \((q;a)\)-tic Veronese↩︎

— The linear system \(\mathrm{S}^a(V^\vee)\) of \((q;a)\)-tic polynomials, being a tensor product of Frobenius twists of the complete linear systems \(\Gamma(\mathbf{P}V, \mathcal{O}_{\mathbf{P}V}(a_j))\), is base point free and so defines a morphism \[\mathrm{Ver}_a \colon \mathbf{P}V \to \mathbf{P}(\mathrm{S}^a(V^\vee)^\vee)\] called the \((q;a)\)-tic Veronese morphism. This description shows that \(\mathrm{Ver}_a\) canonically factors as \[\mathrm{Ver}_a \colon \mathbf{P}V \stackrel{(\mathrm{Ver}_{a_j})}{\longrightarrow} \prod\nolimits_{j \geq 0} \mathbf{P}(\mathop{\mathrm{Sym}}^{a_j}(V^\vee)^\vee) \stackrel{\prod \mathop{\mathrm{Fr}}^j}{\longrightarrow} \prod\nolimits_{j \geq 0} \mathbf{P}(\mathop{\mathrm{Sym}}^{a_j}(V^\vee)^{\vee,[j]}) \stackrel{\mathrm{Seg}}{\longrightarrow} \mathbf{P}(\mathrm{S}^a(V^\vee)^\vee)\] where the first map is the tuple whose \(j\)-th factor is the \(a_j\)-th Veronese embedding; the second map is a product of powers of the \(q\)-power \(\mathbf{k}\)-linear Frobenius morphism, with \(\mathop{\mathrm{Fr}}^j\) acting on the \(j\)-th factors; and the third map is the Segre embedding. This gives the first statement of:

2.4 Example↩︎

— Consider \(a = t^r(1 + t^m)\) for integers \(r \geq 0\) and \(m > 0\), and \(\mathbf{P}V \cong \mathbf{P}^1\). Then the \(a\)-th Veronese is the morphism \(\mathbf{P}^1 \to \mathbf{P}^3\) given by \[\mathrm{Ver}_{t^r(1 + t^m)}(x:y) = (x^{q^r(1 + q^m)}: (x^{q^m} y)^{q^r} : (x y^{q^m})^{q^r} : y^{q^r(1+q^m)}).\] When \(r = 0\), then \(\mathrm{Ver}_{1 + t^m} \colon \mathbf{P}^1 \to \mathbf{P}^3\) is an isomorphism onto its image, providing what might be viewed as a generalization of the twisted cubic, which may be obtained by taking \(q = 2\) and \(m = 1\).

2.5 Parameter space↩︎

— Let \(\mathbf{a}\) be a multi-profile. A parameter space for \((q;\mathbf{a})\)-tic schemes in \(\mathbf{P}V\) is given by the multi-projective space \[(q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \prod\nolimits_{a \in \mathbf{a}} \mathbf{P}\mathrm{S}^a(V^\vee).\] The tautological line subbundles come together to form a tautological \((q;\mathbf{a})\)-tic tensor \[\boldsymbol{\alpha}_{\mathrm{taut}} \colon \bigoplus\nolimits_{a \in \mathbf{a}} \mathop{\mathrm{pr}}_a^*\mathcal{O}_{\mathbf{P}\mathrm{S}^a(V^\vee)}(-1) \to \mathcal{O}_{(q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}} \otimes \mathrm{S}^{\mathbf{a}}(V^\vee)\] which cuts out a tautological family \(\mathcal{X}_{\mathrm{taut}}\) in the product \(\mathbf{P}V \times (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\).

Standard arguments show that the second projection \(\mathcal{X}_{\mathrm{taut}} \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\) is dominant if and only if \(\#\mathbf{a} \leq n\). In this case, as usual, a property is said to hold for a general \((q;\mathbf{a})\)-tic scheme in \(\mathbf{P}V\) if it holds for each fibre of \(\mathcal{X}_{\mathrm{taut}}\) over a nonempty open subset of \((q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\). In the following, call the multi-profile \(\mathbf{a}\) reduced if every \(a \in \mathbf{a}\) is reduced:

It would be interesting to study the discriminant locus \(\Delta\). For \(q\)-bic hypersurfaces as in [qatics-examples][qatics-examples46qbic], \(\Delta\) is cut out by a determinantal equation, similar to the case of quadrics, and carries an intricate stratification corresponding to singularity types of the corresponding hypersurfaces: see [36] for details. It would also be interesting to compute numerical invariants of \(\Delta\), such as its multi-degree, as is done in [38] in the classical case.

3 Linear spaces↩︎

One of the most apparent special properties of a \((q;\mathbf{a})\)-tic complete intersection \(X \subseteq \mathbf{P}V\) is that it contains many more linear spaces than would be expected given its numerical degree. From the point of view of this article, this is because its Fano schemes \(\mathbf{F}_r(X)\)—the Hilbert schemes parameterizing \(r\)-planes contained in \(X\)—inherit a \((q;\mathbf{a})\)-tic structure, resulting in a smaller set of defining equations in the Grassmannian \(\mathbf{G} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{G}(r+1,V)\) of \(r\)-planes in \(\mathbf{P}V\). Writing \(\mathcal{S}\) for the tautological subbundle of rank \(r+1\) on \(\mathbf{G}\), this explicitly means the following:

This provides a lower bound on the dimension of the Fano scheme \(\mathbf{F}_r(X)\) of \(r\)-planes associated with a \((q;\mathbf{a})\)-tic scheme \(X \subseteq \mathbf{P}V\) in an \(n\)-dimensional projective space: If \(\mathbf{F}_r(X) \neq \varnothing\), then \[\dim \mathbf{F}_r(X) \geq \delta(n,\mathbf{a},r) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = (r+1)(n-r) - \sum\nolimits_{a \in \mathbf{a}} \prod\nolimits_{j \geq 0} \binom{a_j + r}{r} \;\;\text{where}\; a = \sum\nolimits_{j \geq 0} a_j t^j.\] Notably, the expected dimension \(\delta(n,\mathbf{a},r)\) of the Fano scheme, depends on \(\mathbf{a}\) but not on \(q\); equivalently, this means that the expected dimension of \(\mathbf{F}_r(X)\) does not directly depend on the degree of \(X\)!

The main result of this section is that \(\mathbf{F}_r(X)\) is of its expected dimension for a general \((q;\mathbf{a})\)-tic scheme \(X\), and will be furthermore smooth provided that the multi-profile \(\mathbf{a}\) is reduced. As is well-known, the statement requires a slight modification when \(X\) is a quadric, in which case the dimension estimate \(\delta(n,\mathbf{a},r)\) is too big when \(r \approx \frac{1}{2}n\). To give a uniform statement, define \[\delta_-(n,\mathbf{a},r) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \min\{\delta(n,\mathbf{a},r), n - 2r - \#\mathbf{a}\}.\] One may verify that, other than in the case \(\mathbf{a} = (2t^k) \cup \mathbf{a}'\) where each \(a \in \mathbf{a}'\) is of the form \(t^m\), \(\delta(n,\mathbf{a},r) \geq 0\) if and only if \(\delta_-(n,\mathbf{a},r) \geq 0\), and similarly for \(> 0\). The statement is now the following:

The proof of 1 occupies the bulk of this section: see, in particular, [fano-theorem-proof] where the intervening statements are put together to complete the argument. Begin with two reductions:

3.1 Reductions↩︎

— First, it suffices to treat the case \(\mathbf{a}\) is reduced. If \(\mathbf{a}\) is nonreduced, let \(\mathbf{a}'\) be the reduced multi-profile obtained by maximally dividing out powers of \(t\) from each profile in \(\mathbf{a}\). Given a \((q;\mathbf{a})\)-tic scheme \(X\), there is a canonical \((q;\mathbf{a}')\)-tic scheme \(X'\) obtained by taking \(q\)-power roots of the appropriate equations. This comes with a closed immersion \(X' \to X\) which is furthermore a universal homeomorphism. This provides a universal homeomorphism \(\mathbf{F}_r(X') \to \mathbf{F}_r(X)\) and so topological properties of the two are the same.

Second, we may assume that \(1 \notin \mathbf{a}\). Otherwise, amongst the defining equations of a \((q;\mathbf{a})\)-tic scheme \(X \subseteq \mathbf{P}V = \mathbf{P}^n\) is a linear one. Eliminating that allows us to view \(X \subseteq \mathbf{P}^{n-1}\). Writing \(\mathbf{a} = (1) \cup \mathbf{a}'\), a direct computation shows that \(\delta_-(n,\mathbf{a},r) = \delta_-(n-1,\mathbf{a}',r)\), and so it suffices to prove 1 for \(X\) viewed as a \((q;\mathbf{a}')\)-tic scheme in \(\mathbf{P}^{n-1}\).

From now on, assume that the profile \(\mathbf{a}\) is reduced and \(1 \notin \mathbf{a}\). The argument looks to lower bound the codimension of the singular locus \(Z_r\) of the second projection from the incidence correspondence \[\mathbf{Inc}_r \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \mathbf{Inc}_{V,r,\mathbf{a}} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \big\{ ([U], [\boldsymbol{\alpha}]) \in \mathbf{G}(r+1,V) \times (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V} : \mathbf{P}U \subseteq X_{\boldsymbol{\alpha}} \big\}\] to the parameter space of \((q;\mathbf{a})\)-tic tensors from [qatic-parameter-space]. Observe that \(\mathop{\mathrm{pr}}_1 \colon \mathbf{Inc}_r \to \mathbf{G}(r+1,V)\) is surjective, with the fibre over a point \([U]\) a multi-projective space given by \[\mathbf{Inc}_{r,[U]} = \prod\nolimits_{a \in \mathbf{a}} \mathbf{P}\mathrm{S}^a(V^\vee)_U\] where \(\mathrm{S}^a(V^\vee)_U \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\ker(\mathrm{S}^a(V^\vee) \to \mathrm{S}^a`(U^\vee))\) is the kernel of the restriction map. This has codimension \(\sum\nolimits_{a \in \mathbf{a}} \prod_{j \geq 0} \binom{r + a_j}{r}\) in the product and so gives:

The first task is to explicitly describe the closed subset \(Z_r \subseteq \mathbf{Inc}_r\) on which \(\mathop{\mathrm{pr}}_2 \colon \mathbf{Inc}_r \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\) is not smooth of expected dimension \(\delta(n,\mathbf{a},r)\): Let \(\mathbf{P}U\) be any \(r\)-plane contained in \(X \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =X_{\boldsymbol{\alpha}}\). Write \(\mathbf{a} = (a_1,\ldots,a_c)\) as a \(c\)-tuple of profiles \(a_i = \sum\nolimits_{j \geq 0} a_{i,j} t^j\), and let \(\alpha_i \in \mathrm{S}^{a_i}(V^\vee)\) be the components of the \((q;\mathbf{a})\)-tic tensor \(\boldsymbol{\alpha}\) defining \(X\). Consider the map \[\begin{align} \rho_{U,f_{\boldsymbol{\alpha}}} \colon U^\vee \otimes (V/U) & \to \bigoplus\nolimits_{i = 1}^c \mathrm{H}^0(\mathbf{P}U, \mathcal{O}_{\mathbf{P}U}(a_i(q))) \\ \xi \otimes \bar{v} & \mapsto ( \xi \cdot \partial_v f_{\alpha_1}\rvert_{\mathbf{P}U}, \ldots, \xi \cdot \partial_v f_{\alpha_c}\rvert_{\mathbf{P}U} ) \end{align}\] which takes a pure tensor \(\xi \otimes \bar{v}\) to the \(c\)-tuple of polynomials whose \(i\)-th entry is \(\xi\) times the directional derivative \(\partial_v f_{\alpha_i}\) of the \(i\)-th equation of \(X\) along any lift \(v \in V\) of \(\bar{v}\), all then restricted to \(\mathbf{P}U\). Since first-order derivatives act linearly through \(q\)-powers, \(\rho_{U,f_{\boldsymbol{\alpha}}}\) in fact takes values within the subspace of \((q;\mathbf{a})\)-tic polynomials. More precisely:

If \(\mathbf{P}U\) were contained in the smooth locus of \(X\), then the tangent space to \(\mathbf{F}_r(X)\) at the point \([\mathbf{P}U]\) is given by the space of sections the normal bundle of \(\mathbf{P}U \subseteq X\). The normal bundle sequence, 1, and injectivity of \(\mathop{\mathrm{mult}}\) together given canonical identifications \[\mathrm{H}^0(\mathbf{P}U, \mathcal{N}_{\mathbf{P}U/X}) \cong \ker\rho_{U,f_{\boldsymbol{\alpha}}} \cong \ker\rho_{U,\boldsymbol{\alpha}}.\] Therefore \(\mathbf{F}_r(X)\) is smooth of dimension \(\delta(n,\mathbf{a},r)\) at the point \([\mathbf{P}U]\) if and only if the map \(\rho_{U,\boldsymbol{\alpha}}\) is surjective. This property characterizes the complement of \(Z_r\) in general:

This description of \(Z_r\) is homogeneous in \([U]\), making it possible to fix a subspace \(U\) and to simply estimate the codimension of the inclusion of fibres \(Z_{r,[U]} \subset \mathbf{Inc}_{r,[U]}\). Proceed by additionally parameterizing a bound on the image of \(\rho_{U,\boldsymbol{\alpha}}\): Let \(\mathbf{H}\) be the projective space of hyperplanes in \(\mathrm{S}^{\mathbf{a}}(U^\vee)\), and consider the correspondence \[Z_{r,[U]}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \big\{([\boldsymbol{\alpha}],[\varphi]) \in \mathbf{Inc}_{r,[U]} \times \mathbf{H} : \varphi \circ \rho_{U,\boldsymbol{\alpha}} \colon U^\vee \otimes (V/U) \to \mathbf{k} \;\text{is zero} \big\}\] parameterizing \(X_{\boldsymbol{\alpha}}\) containing \(\mathbf{P}U\) and a hyperplane in \(\mathrm{S}^{\mathbf{a}}(U^\vee)\) containing the image of \(\rho_{U,\boldsymbol{\alpha}}\). Projection to \(\mathbf{Inc}_{r,[U]}\) maps this onto \(Z_{r,[U]}\). The fibre of projection to \(\mathbf{H}\) over a point \(\varphi \colon \mathrm{S}^{\mathbf{a}}(U^\vee) \to \mathbf{k}\) is a product of projective spaces on the kernel of the map \[\Phi \colon \mathrm{S}^{\mathbf{a}}(V^\vee)_U \to \mathop{\mathrm{Hom}}(U^\vee \otimes (V/U), \mathbf{k}) \quad\quad \boldsymbol{\alpha} \mapsto \varphi \circ \rho_{U,\boldsymbol{\alpha}}.\] Writing \(\mathbf{a} - 1 = (a_1 - 1, \ldots, a_c - 1)\), the following gives a stratification of the image \(Z_{r,[U]}'\) in \(\mathbf{H}\) over which the fibres have the same dimension:

To relate this with the codimension of \(Z_{r,[U]}\) in \(\mathbf{Inc}_{r,[U]}\), it remains to bound the dimension of \(\mathbf{H}_k\):

Non-smooth points of \(\mathbf{F}_{r-1}(X)\) often contribute to non-smooth points of \(\mathbf{F}_r(X)\), producing components of \(Z_r\) that are too large. Writing \(\Delta_r\) for the image of \(Z_r\) under \(\mathop{\mathrm{pr}}_2 \colon \mathbf{Inc}_r \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\), the following statement says that members of \(Z_r^\circ \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =Z_r \setminus \mathop{\mathrm{pr}}_2^{-1}(\Delta_{r-1})\) are parameterized by the piece of highest codimension from 1:

3.2 ↩︎

— It remains to put everything together to prove 1:

For [fano-theorem46empty], if \(\delta_-(n,\mathbf{a},r) < 0\) and \(\mathbf{a} \neq (2)\), then \(\mathop{\mathrm{pr}}_2 \colon \mathbf{Inc}_r \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\) cannot be dominant by the dimension computation in 1, and so \(\mathbf{F}_r(X) = \varnothing\) for general \(X\). The case \(\mathbf{a} = (2)\) is well-known.

For [fano-theorem46smooth], if \(\delta_-(n,\mathbf{a},r) \geq 0\), then it follows from 1 that the intersection of \(Z_r \setminus \mathop{\mathrm{pr}}_2^{-1}(\Delta_{r-1})\) with \(\mathbf{Inc}_{r,[U]}\) is contained in the image of \(Z_{r,[U]}' \times_{\mathbf{H}} \mathbf{H}_r\) for each \([U] \in \mathbf{G}(r+1,V)\). The argument of 1 then implies that the codimension of its closure in \(\mathbf{Inc}_r\) is at least \(\delta_-(n,\mathbf{a},r) + 1\), and so \[\dim\big(\overline{Z_r \setminus \mathop{\mathrm{pr}}_2^{-1}(\Delta_{r-1})}\big) \leq \dim\mathbf{Inc}_r - \delta_-(n,\mathbf{a},r) - 1 \leq \dim\,(q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V} - 1.\] Therefore \(\Delta_r \setminus \Delta_{r-1}\) is not all of \((q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\). Induction on \(r\)—the base case with \(r = 0\) is the statement 1 that the general \((q;\mathbf{a})\)-tic scheme is smooth—then shows that \(\Delta_r\) is a proper closed subset of \((q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\), meaning that \(\mathbf{F}_r(X)\) is smooth of the expected dimension for general \(X\) by 1.

For [fano-theorem46connected], when \(\delta_-(n, \mathbf{a}, r) > 0\), consider the Stein factorization \[\mathop{\mathrm{pr}}_2 \colon \mathbf{Inc}_r \to \mathbf{Inc}_r' \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\] of the second projection. Then \(Z_r\) contains the preimage of the branch locus of \(\mathbf{Inc}_r' \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\). Since \(Z_r\) has codimension at least \(2\) in \(\mathbf{Inc}_r\) by 1, purity of the branch locus, as in [37, p. 0BMB], implies that \(\mathbf{Inc}_r' \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}V}\) is finite étale; the target is a multi-projective space and so it is simply connected, thus this is an isomorphism, and properties of the Stein factorization imply that \(\mathop{\mathrm{pr}}_2\) has connected fibres. In other words, \(\mathbf{F}_r(X)\) is connected for every \(X\). 0◻

When \(\mathbf{F}_r(X)\) is of its expected dimension \(\delta(n,\mathbf{a},r)\), 1 shows that it is cut out in \(\mathbf{G}\) by a regular section of \(\mathrm{S}^{\mathbf{a}}(\mathcal{S}^\vee)\). Various simple numerical invariants of the Fano scheme may then be determined via Schubert calculus, as is done in [14] for classical complete intersections, and [9] and [28] in the \(q\)-bic case. For now, record the fact that the dualizing sheaf \(\omega_{\mathbf{F}_r(X)}\) is a power of the Plücker line bundle \(\mathcal{O}_{\mathbf{F}_r(X)}(1)\):

3.3 \(r\)-planes through a point↩︎

— Let \(X \subseteq \mathbf{P}V\) be a \((q;\mathbf{a})\)-tic scheme and consider the scheme \[\mathbf{F}_r(X,x) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{[\mathbf{P}U] \in \mathbf{F}_r(X) : x \in \mathbf{P}U \subseteq X\}\] parameterizing \(r\)-planes in \(X\) through a given closed point \(x\). As usual, this may be canonically identified as the Fano scheme of \((r-1)\)-planes of a scheme \(X_{1,x} \subseteq \mathbf{P}(V/L)\), where \(L\) is the \(1\)-dimensional space underlying \(x\) and \(X_{1,x} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{F}_1(X,x)\) is the scheme of lines in \(X\) through \(x\). A classical fact, see [39] for example, is that if \(X \subseteq \mathbf{P}V\) is a scheme defined by equations of multi-degree \(\mathbf{d} = (d_1,d_2,\ldots,d_c)\), then \(X_{1,x} \subseteq \mathbf{P}(V/L)\) is defined by equations of multi-degree \[\mathbf{d}_1 \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = ( d_1, d_1 - 1, \ldots, 2, 1;\, d_2, d_2 - 1, \ldots, 2, 1; \ldots;\, d_c, d_c - 1, \ldots, 2, 1).\] A generalization of this to \((q;\mathbf{a})\)-tic schemes is as follows:

Writing \(\mathbf{F}_r(X,x) \cong \mathbf{F}_{r-1}(X_{1,x})\) and combining 1 with 1 and 1 shows that a \((q;\mathbf{a})\)-tic scheme is covered by \(r\)-planes as soon as \(\delta_-(n-1,\mathbf{a}_1,r-1) \geq 0\). A simple criterion for \(\delta(n-1,\mathbf{a}_1,r-1) \geq 0\) may be obtained by using the identity \[1 + \sum\nolimits_{0 \prec b \preceq a} \prod\nolimits_{j \geq 0} \binom{b_j + r - 1}{r - 1} = \prod\nolimits_{j \geq 0} \Big(\sum\nolimits_{b_j = 0}^{a_j} \binom{b_j + r - 1}{r - 1}\Big) = \prod\nolimits_{j \geq 0} \binom{a_j + r}{r}.\]

A more global version of 1 will be given in §5 and will feature in the unirationality construction in Theorem 2. The next section prepares for this by clarifying what a family of \((q;\mathbf{a})\)-tic schemes ought to be and by developing some tools for manipulating such families.

4 Families↩︎

Consider a family \(\mathcal{X} \to S\) of complete intersections in a projective bundle \(\pi \colon \mathbf{P}\mathcal{V} \to S\) over the field \(\mathbf{k}\) which is cut out by a regular section \(\sigma \colon \mathcal{O}_{\mathbf{P}\mathcal{V}} \to \mathcal{E}\) of a finite locally free \(\mathcal{O}_{\mathbf{P}\mathcal{V}}\)-module. What additional structure should be required to elevate \(\mathcal{X}\) into a family of \((q;\mathbf{a})\)-tic complete intersections? As a minimum, each fibre \(\mathcal{X}_s\) ought to be a \((q;\mathbf{a})\)-tic complete intersection in \(\mathbf{P}\mathcal{V}_s\), meaning as in [qatics-definition] that the equations \(\sigma_s\) are induced by a \((q;\mathbf{a})\)-tic tensor \(\boldsymbol{\alpha}_s\). This tensor is a crucial part of the structure that defines a \((q;\mathbf{a})\)-tic complete intersection, so one ought to ask that the \(\boldsymbol{\alpha}_s\) vary continuously across the family. The following examples illustrate some of the subtleties involved:

Excising the origin from the base ensures that a tensor does glue together to become a section of a non-split vector bundle:

Toward a definition of a family of \((q;\mathbf{a})\)-tic schemes, suppose one is given a section of the form \[\boldsymbol{\alpha} \colon \mathcal{O}_S \to \bigoplus\nolimits_{a \in \mathbf{a}} \mathrm{S}^a(\mathcal{V}^\vee).\] Adjunction along \(\pi \colon \mathbf{P}\mathcal{V} \to S\) together with the relative evaluation maps induce a canonical map \[\sigma \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\varepsilon \circ \pi^*\boldsymbol{\alpha} \colon \mathcal{O}_{\mathbf{P}\mathcal{V}} \to \bigoplus\nolimits_{a \in \mathbf{a}} \pi^*\mathrm{S}^a(\mathcal{V}^\vee) \to \bigoplus\nolimits_{a \in \mathbf{a}} \mathcal{O}_\pi(a(q))\] whose zero locus \(\mathcal{X}\) ought to be called a family of \((q;\mathbf{a})\)-tic schemes over \(S\). However, there are families like those in 1 whose defining section takes values in a vector bundle which is only locally of this form. Additional data is therefore necessary to globalize this construction. The solution taken here is to require a map \(\varepsilon\) globalizing the evaluation map.

4.1 Definitions↩︎

— A family of \((q;\mathbf{a})\)-tic tensors valued in a finite locally free \(\mathcal{O}_S\)-module \(\mathcal{V}\) is a section \(\boldsymbol{\alpha} \colon \mathcal{O}_S \to \mathcal{A}\), where \(\mathcal{A}\) is locally on \(S\) isomorphic to \[\bigoplus\nolimits_{a \in \mathbf{a}} \mathrm{S}^a(\mathcal{V}^\vee).\] A family of \((q;\mathbf{a})\)-tic schemes in a projective bundle \(\pi \colon \mathbf{P}\mathcal{V} \to S\) consists of the data of

• a closed subscheme \(\mathcal{X} \subseteq \mathbf{P}\mathcal{V}\) cut out by a section \(\sigma \colon \mathcal{O}_{\mathbf{P}\mathcal{V}} \to \mathcal{E}\) of a vector bundle;

• a family of \((q;\mathbf{a})\)-tic tensors \(\boldsymbol{\alpha} \colon \mathcal{O}_S \to \mathcal{A}\); and

• a surjective morphism \(\varepsilon \colon \pi^*\mathcal{A} \to \mathcal{E}\) of \(\mathcal{O}_{\mathbf{P}\mathcal{V}}\)-modules.

These data are subject to the conditions that:

1. there is a factorization \(\sigma = \varepsilon \circ \pi^*\boldsymbol{\alpha} \colon \mathcal{O}_{\mathbf{P}\mathcal{V}} \to \pi^*\mathcal{A} \to \mathcal{E}\); and

2. there exists an open cover \(S = \bigcup\nolimits_{i \in I} U_i\) on which the morphism \(\varepsilon \colon \pi^*\mathcal{A} \to \mathcal{E}\) is isomorphic to the canonical map induced by evaluation along \(\pi\): \[\mathop{\mathrm{ev}}_\pi \colon \bigoplus\nolimits_{a \in \mathbf{a}} \pi^*\mathrm{S}^a(\mathcal{V}^\vee) \to \bigoplus\nolimits_{a \in \mathbf{a}} \mathcal{O}_\pi(a(q)).\]

When \(\sigma\) is a regular section making \(\mathcal{X} \to S\) fibrewise a complete intersection, furthermore call the triple \((\mathcal{X}, \boldsymbol{\alpha}, \varepsilon)\) a family of \((q;\mathbf{a})\)-tic complete intersections. The family \(\mathcal{X} \to S\) is often referred to as the family of \((q;\mathbf{a})\)-tic schemes or complete intersections, leaving the \(\boldsymbol{\alpha}\) and \(\varepsilon\) implicit.

4.2 Example↩︎

— Since \((q;\mathbf{a})\)-tic structures are inherited upon passing to linear sections by 1, families of linear sections of \((q;\mathbf{a})\)-tic schemes provide a source of non-trivial examples. To give a simple illustration, consider linear projection of a projective space \(\mathbf{P}V\) centred along a subspace \(\mathbf{P}U\). This provides a rational map to \(\mathbf{P}(V/U)\) which is resolved on the blowup \(b \colon \widetilde{\mathbf{P}} V \to \mathbf{P}V\) centred along \(\mathbf{P}U\); as usual, \(b\) exhibits \(\widetilde{\mathbf{P}} V\) as the projective bundle over \(\mathbf{P}(V/U)\) whose underlying vector bundle \(\mathcal{V}\) canonically arises via the diagram of short exact sequences \[\begin{tikzcd} 0 \rar & \mathcal{O}_{\mathbf{P}(V/U)} \otimes U \rar \dar[equal] & \mathcal{V} \rar \dar & \mathcal{O}_{\mathbf{P}(V/U)}(-1) \rar \dar & 0 \\ 0 \rar & \mathcal{O}_{\mathbf{P}(V/U)} \otimes U \rar & \mathcal{O}_{\mathbf{P}(V/U)} \otimes V \rar & \mathcal{O}_{\mathbf{P}(V/U)} \otimes V/U \rar & 0\makebox[0pt][l]{\,.} \end{tikzcd}\] If \(X \subseteq \mathbf{P}V\) is a \((q;\mathbf{a})\)-tic scheme not contained in \(\mathbf{P}U\), then its total transform \(\mathcal{X} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =b^{-1}(X)\) in \(\widetilde{\mathbf{P}} V\) is a family of \((q;\mathbf{a})\)-tic schemes over \(S \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{P}(V/U)\): Dualizing the inclusion \(\mathcal{V} \subseteq \mathcal{O}_{\mathbf{P}(V/U)} \otimes V\) and applying the tensor functor \(\mathrm{S}^{\mathbf{a}}\) provides a restriction map \[\mathcal{O}_{\mathbf{P}(V/U)} \otimes \mathrm{S}^{\mathbf{a}}(V^\vee) \to \mathrm{S}^{\mathbf{a}}(\mathcal{V}^\vee).\] Mapping a \((q;\mathbf{a})\)-tic tensor defining \(X\) along this provides a family of \((q;\mathbf{a})\)-tic tensors \(\boldsymbol{\alpha}\) defining \(\mathcal{X}\).

4.3 ↩︎

— If, furthermore, \(\mathbf{P}U \subseteq X\), then its strict transform \(\widetilde{X}\) in \(\widetilde{\mathbf{P}} V\) generally does not appear to carry useful additional structure from the point of view of this article. For instance, consider the smooth \(q\)-bic surface \[X \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \set{ (x_0:x_1:x_2:x_3) \in \mathbf{P}^3 : x_0^q x_1 + x_0 x_1^q + x_2^q x_3 + x_2 x_3^q = 0 }.\] Projection from the line \(\mathbf{P}U = (0:x_1:0:x_3)\) exhibits the strict transform \(\widetilde{X}\) as a family of degree \(q\) plane curves, the general fibre of which is isomorphic to \[C \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\set{(y_0:y_1:y_2) \in \mathbf{P}^2 : y_0^q + y_1y_2^{q-1} = 0},\] see [40, p. 2.5.3] for instance. In general, this equation does not belong to any proper subspace of \(\mathop{\mathrm{Sym}}^q(\mathbf{k}^{\oplus 3})\) of the form \(\mathrm{S}^a(\mathbf{k}^{\oplus 3})\) for \(a \in \mathbf{Prfl}\) of numerical degree \(q\). This perhaps suggests that the class of \((q;\mathbf{a})\)-tic schemes is not sufficiently flexible. It would be useful to develop methods to handle a larger class of schemes that includes irreducible components of \((q;\mathbf{a})\)-tic schemes.

4.4 Transition functions↩︎

— Injectivity of multiplication maps associated with profiles from [qatics-profiles] together with condition [families-complete-intersections][families-complete-intersections46eta] implies that the map \(\pi_*\varepsilon \colon \mathcal{A} \to \pi_*\mathcal{E}\) is an injection which locally on \(S\) is isomorphic to the inclusion of \((q;a)\)-tic polynomials amongst all polynomials of degree \(a(q)\), with \(a\) ranging over profiles in \(\mathbf{a}\). This endows \(\mathcal{A}\) with some additional structure: Choose an open covering \(S = \bigcup\nolimits_{i \in I} U_i\) and trivializations of \(\mathcal{A}\) and \(\pi_*\mathcal{E}\) so that \(\pi_*\varepsilon\rvert_{U_i}\) is the identified with the inclusion \[\bigoplus\nolimits_{a \in \mathbf{a}} \mathrm{S}^a(\mathcal{V}^\vee)\rvert_{U_i} \subseteq \bigoplus\nolimits_{a \in \mathbf{a}} \mathop{\mathrm{Sym}}^{a(q)}(\mathcal{V}^\vee)\rvert_{U_i}.\] For each pair of indices \(i,j \in I\), let \(\psi_{i,j}\) and \(\varphi_{i,j}\) be the transition functions of \(\mathcal{A}\) and \(\pi_*\mathcal{E}\) with these trivializations over \(U_{i,j} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =U_i \cap U_j\). Since \(\mathcal{E}\) is locally a sum of the \(\mathcal{O}_\pi(a(q))\), the \(\varphi_{i,j}\) are induced by multiplication of polynomials; in particular, for \(a, b \in \mathbf{a}\), the \((a,b)\)-component \[(\varphi_{i,j})_{a,b} \colon \mathop{\mathrm{Sym}}^{a(q)}(\mathcal{V}^\vee)\rvert_{U_{i,j}} \to \mathop{\mathrm{Sym}}^{b(q)}(\mathcal{V}^\vee)\rvert_{U_{i,j}}\] is induced by multiplication by a polynomial \(f_{i,j,a,b}\) of degree \(b(q) - a(q)\). Since \(\mathcal{A}\) is identified as a subbundle of \(\pi_*\mathcal{E}\), this means that the map \[(\psi_{i,j})_{a,b} \colon \mathrm{S}^a(\mathcal{V}^\vee)\rvert_{U_{i,j}} \to \mathrm{S}^b(\mathcal{V}^\vee)\rvert_{U_{i,j}}\] is also induced by multiplication by the same polynomial \(f_{i,j,a,b}\). Keeping track of profiles shows that whether or not \((\psi_{i,j})_{a,b}\) must vanish is related to the composite partial ordering \(\rightsquigarrow\) from 1:

As a simple though useful consequence, this provides a \((q;\mathbf{a})\)-tic tensor \(\boldsymbol{\alpha} \colon \mathcal{O}_S \to \mathcal{A}\) with a canonical decomposition into types. Namely, partition the collection \(\mathbf{a}\) of profiles into \[\mathbf{a}_{\mathrm{lin}} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{a \in \mathbf{a} : a(t) = 1\}, \;\; \mathbf{a}_{\mathrm{pow}} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{a \in \mathbf{a} : a(0) = 0\},\;\; \mathbf{a}_{\mathrm{nlr}} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{a \in \mathbf{a} : a(t) \neq 1 \;\text{and}\; a(0) = 0\}\] those that are linear, those that are nonreduced, and those that are non-linear and reduced. The linear and nonreduced components of \(\mathcal{A}\) give two canonical quotients:

Dually, 1 implies that equations whose profile is maximal for \(\rightsquigarrow\) in \(\mathbf{a}\) fit together into a subbundle of \(\mathcal{A}\). One such class of maximal profiles are those \(a \in \mathbf{a}\) with maximal coefficient sum \(a(1)\): see [qatics-ordering-properties][qatics-ordering-properties46coefficient-sums]. This is most useful when restricted to \(\mathcal{A}_{\mathrm{nlr}} \neq 0\):

The canonical type decomposition from 1 of a \((q;\mathbf{a})\)-tic tensor allows one to sometimes perform certain simplifications to the equations defining the family \(\mathcal{X}\). For instance—a construction which is of course much more generally applicable to any family of projective schemes—the linear equations may be used to cut out a projective subbundle containing \(\mathcal{X}\), thereby reducing the number of equations to keep track of. Precisely:

Perhaps a more interesting simplification is possible when the collection of profiles \(\mathbf{a} = \mathbf{a}_{\mathrm{pow}}\) consists only of nonreduced profiles, meaning that each equation of \(\mathcal{X}\) over \(S\) is, geometrically, a \(q\)-power. Upon adjoining suitable roots of the coefficients, it is possible to take \(q\)-th roots of all the equations of \(\mathcal{X}\) to obtain a scheme \(\mathcal{X}'\) with multi-profile \(\mathbf{a}/t \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =(a/t : a \in \mathbf{a})\):

4.5 Planing↩︎

— An \(r\)-planing of a family \(\mathcal{X}\) of \((q;\mathbf{a})\)-tic schemes in \(\pi \colon \mathbf{P}\mathcal{V} \to S\) is the data of a closed subscheme \(\mathcal{P} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{P}\mathcal{U} \subseteq \mathcal{X}\) corresponding to a subbundle \(\mathcal{U} \subseteq \mathcal{V}\) of rank \(r+1\); in other words, \(\mathcal{P}\) is a family of \(r\)-planes contained in \(\mathcal{X}\). The pair \(\mathcal{P} \subseteq \mathcal{X}\) is referred to as a family of \(r\)-planed \((q;\mathbf{a})\)-tic schemes over the base scheme \(S\). Observe that an \(r\)-planing \(\mathcal{P}\) of \(\mathcal{X}\) may be transformed into an \(r'\)-planing for any \(0 \leq r' \leq r\) upon base change to the Grassmannian bundle \(\mathbf{G}(r'+1,\mathcal{U}) \to S\) and taking the projective bundle on the tautological subbundle of rank \(r'+1\).

4.6 Classifying maps↩︎

— Let \(\mathcal{P} \subseteq \mathcal{X}\) be an \(r\)-planed family of \((q;\mathbf{a})\)-tic schemes in a \(\mathbf{P}^n\)-bundle over a base \(S\). At least on open subsets of the base \(S\), the family \(\mathcal{X}\) and the pair \(\mathcal{P} \subseteq \mathcal{X}\) induce classifying maps to the parameter spaces \[\begin{align} (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}^n} & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \prod\nolimits_{a \in \mathbf{a}} \mathbf{P}\mathrm{S}^a(\mathbf{k}^{\oplus n+1})\;\text{and} \\ \mathbf{Inc}_{n,r,\mathbf{a}} & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \Set{ ([U], [\boldsymbol{\alpha}]) \in \mathbf{G}(r+1,n+1) \times (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}^n} : \mathbf{P}U \subseteq X_{\boldsymbol{\alpha}}}, \end{align}\] encountered already in [qatic-parameter-space] and 1. To describe these, it is convenient to assume that \(S\) is integral, so that the classifying maps may be seen as rational maps from \(S\) to one of the two parameter spaces; often, such a rational map will stand in for a choice of classifying map constructed below.

Assume henceforth that the base scheme \(S\) is integral. For a nonempty open subscheme \(S^\circ \subseteq S\), write \(\mathcal{V}^\circ\), \(\mathcal{P}^\circ\), and \(\mathcal{X}^\circ\) for the restrictions of \(\mathcal{V}\), \(\mathcal{P}\), and \(\mathcal{X}\) over \(S^\circ\). Pick \(S^\circ\) over which \(\mathcal{V}^\circ\) is trivial and \(\varepsilon\) may be identified with the evaluation map as in [families-complete-intersections][families-complete-intersections46eta]. Fixing suitable trivializations identifies \(\mathcal{X}^\circ\) as a \((q;\mathbf{a})\)-tic scheme in the projective space \(\mathbf{P}\mathcal{V}^\circ \cong \mathbf{P}^n \times S^\circ\) defined by a \((q;\mathbf{a})\)-tic tensor \[\boldsymbol{\alpha}^\circ \colon \mathcal{O}_{S^\circ} \to \bigoplus\nolimits_{a \in \mathbf{a}} \mathrm{S}^a(\mathcal{V}^{\circ,\vee}).\] This tensor defines a morphism \(S^\circ \to (q;\mathbf{a})\operatorname{\boldsymbol{\!}-tics}_{\mathbf{P}^n}\) and, together with \(\mathcal{P}^\circ\), a morphism \(S^\circ \to \mathbf{Inc}_{n,r,\mathbf{a}}\). These are the classifying morphisms for the families \(\mathcal{X}^\circ\) and \(\mathcal{P}^\circ \subseteq \mathcal{X}^\circ\), respectively.

4.7 Generic families↩︎

— Different choices of trivialization produce classifying morphisms which differ by automorphisms of the parameter spaces, so dominance of a classifying map does not depend on any of the choices above; after all, dominance means informally that the family contains the general member of the parameter space in question. By way of terminology, call the family \(\mathcal{P} \subseteq \mathcal{X}\) generic if the associated classifying map \(S \dashrightarrow \mathbf{Inc}_{n,r,\mathbf{a}}\) is dominant.

Genericity propagates along many constructions. As a first example, consider a family \(\mathcal{P} \subseteq \mathcal{X}\) of \(r\)-planed \((q;\mathbf{a})\)-tic schemes in which some of the equations of \(\mathcal{X}\) are linear: that is, \(\mathbf{a}_{\mathrm{lin}} \neq \varnothing\) in the notation of 1. Then \(\mathcal{P} \subseteq \mathcal{X}\) may be considered as a family in the smaller projective bundle \(\mathbf{P}\mathcal{V}'\) cut out by those linear equations, as in 1. Since an equation in a projective subspace may be extended—in many ways!—to an equation in a larger ambient projective space, it is easy to convince oneself that if \(\mathcal{P} \subseteq \mathcal{X}\) is generic as a family in \(\mathbf{P}\mathcal{V}\), then it remains generic as a family in \(\mathbf{P}\mathcal{V}'\). A more careful argument is given in the following:

5 Highly tangent lines↩︎

Lines tangent to order \(\geq k\) to a projective scheme \(X \subseteq \mathbf{P}^n\) may be parameterized by a scheme of the form \[\mathbf{Tan}_k(X) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \{(x,[\ell]) \in X \times \mathbf{G}(2,n+1) : \mathop{\mathrm{mult}}_x(\ell \cap X) \geq k\}.\] Familiar cases include: When \(k = 1\), this is the restriction to \(X\) of the variety of pointed lines in \(\mathbf{P}^n\), and is the projective bundle on \(\mathcal{T} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathcal{T}_{\mathbf{P}^n}(-1)\rvert_X\). When \(k = 2\), this is the projectivized tangent bundle of \(X\). When \(X\) is a general hypersurface of degree \(d\), it is well-known that the general fibre of the projection \(\mathbf{Tan}_k(X) \to X\) to the \(x\)-coordinate is a complete intersection of type \((k-1,k-2,\ldots,2,1)\) in an \((n-1)\)-dimensional projective space.

If \(X\) is a general complete intersection of codimension \(c \geq 2\), \(\mathbf{Tan}_k(X)\) usually consists of several components and will not fibre in complete intersections over \(X\). Nonetheless, upon writing \(X\) as an intersection \(H_1 \cap \cdots \cap H_c\) of hypersurfaces \(H_i\) of degree \(d_i\), each of the schemes \[\mathbf{Tan}_k(X; H_i) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \{(x,\ell) : \mathop{\mathrm{mult}}_x(\ell \cap H_i) \geq k\;\text{and}\; \ell \subset H_j\;\text{for \(1 \leq j \leq c\) and \(j \neq i\)}\},\] parameterizing \(k\)-fold tangent lines to \(H_i\) and pointed lines in the remaining \(H_j\), are distinguished components of \(\mathbf{Tan}_k(X)\) which project onto \(X\) with complete intersection general fibres. The aim of this section is to make sense of this construction and structure for a family of \((q;\mathbf{a})\)-tic schemes.

Rather than discussing \(\mathbf{Tan}_k(X)\) for general \(k\), this section focuses on the case of particular relevance, namely, \(k = d - 1\) where \(d \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\deg X\): \[\mathbf{PenTa}(X) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{Tan}_{d-1}(X) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\{(x,[\ell]) \in X \times \mathbf{G}(2,n+1) : \mathop{\mathrm{mult}}_x(\ell \cap X) \geq d-1\}.\] Lines parameterized by \(\mathbf{PenTa}(X)\) are called penultimate tangents. In words, this scheme of penultimate tangents parameterizes pointed lines \(x \in \ell \subset \mathbf{P}^n\) such that either \(\ell\) is contained in \(X\) or else their intersection, viewed as a Cartier divisor on \(\ell\), is of the form \(\ell \cap X = (d-1)x + x'\) for a residual point \(x' \in X\). The interest in this case is that extracting the residual point often provides a rational map \(\mathop{\mathrm{res}}\colon \mathbf{PenTa}(X) \dashrightarrow X\), which will be studied in §6.

5.1 Local situation↩︎

— When \(S = \mathop{\mathrm{Spec}}\mathbf{k}\) and \(X \subseteq \mathbf{P}^n\) is a \((q;\mathbf{a})\)-tic scheme, the situation is easy to describe explicitly. With the notation in the proof of 1, the scheme \(X_1\) of pointed lines in \(X\) is defined over \(\mathrm{D}(x_n) \subseteq \mathbf{P}^n\) as \[X_1\rvert_{\mathrm{D}(x_n)} = \{ (\mathbf{x}, \mathbf{y}) \in \mathrm{D}(x_n) \times \mathbf{P}^{n-1} : f_{a,b}(\mathbf{x};\mathbf{y}) = 0 \;\text{for}\; a \in \mathbf{a} \;\text{and}\; 0 \preceq b \preceq a \}.\] A point \((\mathbf{x},\mathbf{y}) \in \mathrm{D}(x_n) \times \mathbf{P}^{n-1}\) corresponds to a parameterized line \((\xi:\eta) \mapsto \xi\mathbf{x} + \eta\mathbf{y}\), so the point \(\mathbf{x}\) is the image of \((1:0)\). Writing \(H \subseteq \mathbf{P}^n\) for the hypersurface cut out by a \((q;a)\)-tic defining polynomial \(f_a\), the scheme \(\mathbf{Tan}_k(H)\) of pointed lines tangent to \(H\) to order \(\geq k\) is defined by \(f_{a,b}(\mathbf{x};\mathbf{y}) = 0\) for \(0 \preceq b \preceq a\) satisfying \(b(q) \leq k\). Therefore, over \(X \cap \mathrm{D}(x_n)\), the scheme \(\mathbf{PenTa}(X;H)\) of penultimate tangents may be obtained from \(X_1\) by omitting the two equations \(f_{a,a}(\mathbf{x};\mathbf{y}) = f_{a,a-1}(\mathbf{x};\mathbf{y}) = 0\), presenting it as a family of \((q;\mathbf{a}')\)-tic where \(\mathbf{a}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{a}_1 \setminus (a,a-1)\); this is most useful when \(a - 1\) is a nonzero profile, equivalently when the profile \(a\) is nonlinear and reduced.

The next few paragraphs make sense of these observations in families.

5.2 Pointed lines↩︎

— Given a projective bundle \(\pi \colon \mathbf{P}\mathcal{V} \to S\), write \(\mathbf{L}\mathcal{V}\) for its space of pointed lines: namely, the incidence correspondence between it and its relative Grassmannian \(\mathbf{G}\mathcal{V}\) of lines. Projection to the line exhibits \(\mathbf{L}\mathcal{V}\) as the universal line over \(\mathbf{G}\mathcal{V}\), whereas projection \(\mathop{\mathrm{pr}}_x \colon \mathbf{L}\mathcal{V} \to \mathbf{P}\mathcal{V}\) to the point identifies \(\mathbf{L}\mathcal{V}\) as the projective bundle on the relative tangent bundle \(\mathcal{T}_\pi \otimes \mathcal{O}_\pi(-1)\) of \(\pi \colon \mathbf{P}\mathcal{V} \to S\). These moduli descriptions imply that the tautological bundles of \(\mathbf{L}\mathcal{V}\) fit into a canonical short exact sequence \[0 \to \mathcal{O}_{\mathop{\mathrm{pr}}_x}(1) \to \mathcal{S}^\vee \to \mathop{\mathrm{pr}}_x^*\mathcal{O}_\pi(1) \to 0\] where \(\mathcal{S}\) is the subbundle of rank \(2\). Pushing along \(\mathop{\mathrm{pr}}_x\) yields the relative dual Euler sequence \[0 \to \Omega^1_\pi \otimes \mathcal{O}_\pi(1) \to \pi^*\mathcal{V}^\vee \to \mathcal{O}_\pi(1) \to 0.\]

Consider now a family \(\mathcal{X}\) of \((q;\mathbf{a})\)-tic schemes in \(\pi \colon \mathbf{P}\mathcal{V} \to S\). Its space of pointed lines \[\mathcal{X}_1 \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \{(x, \ell) \in \mathcal{X} \times_S \mathbf{F}_1(\mathcal{X}/S) : x \in \ell\} \hookrightarrow \mathbf{L}\mathcal{V}\rvert_{\mathcal{X}}\] naturally embeds into the projective bundle on \(\mathcal{T} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathcal{T}_\pi \otimes \mathcal{O}_\pi(-1)\rvert_{\mathcal{X}}\) in which it has a \((q;\mathbf{a}_1)\)-tic structure, globalizing 1:

In light of the discussion in [tlines-explicit-construction], it will be useful to describe how to access, within the bundle \(\mathcal{E}_1\) of equations for \(\mathcal{X}_1\) in \(\mathbf{P}\mathcal{T}\) constructed in 1, those of degree \(a(q)\) and \(a(q) - 1\), at least when \(\mathcal{X}\) is a family of \((q;a)\)-tic hypersurfaces with \(a\) nonlinear and reduced:

5.3 Penultimate tangents↩︎

— Let \(\mathcal{X}\) be a family of \((q;\mathbf{a})\)-tic schemes in \(\pi \colon \mathbf{P}\mathcal{V} \to S\), and let \(\mathcal{X}_1\) be its scheme of pointed lines equipped with its structure as a family of \((q;\mathbf{a}_1)\)-tic schemes in \(\rho \colon \mathbf{P}\mathcal{T} \to \mathcal{X}\) as constructed in 1. Assume that \(\mathbf{a}_{\mathrm{nlr}} \neq \varnothing\) and choose a subbundle \(\mathrm{S}^a(\mathcal{V}^\vee) \otimes \mathcal{M} \subseteq \mathcal{A}_{\mathrm{nlr}}\) as in 1. To perform the construction [tlines-explicit-construction] in families, view the projective bundle \(\mu \colon \mathbf{P}\mathcal{M} \to S\) as the linear system parameterizing \((q;a)\)-tic hypersurfaces containing \(\mathcal{X}\) corresponding to the equations of \(\mathrm{S}^a(\mathcal{V}^\vee) \otimes \mathcal{M}\). On the product \(\mathbf{P}\mathcal{T} \times_S \mathbf{P}\mathcal{M}\), the chosen subbundle of \(\mathcal{A}\) together with the tautological line subbundle \(\mathcal{O}_\mu(-1)\), and the computation of 1 distinguish subbundles \[\mathop{\mathrm{pr}}_1^*\mathcal{E}_1 \supseteq \ker\big(\mathrm{S}^a(\mathcal{S}^\vee) \to \rho^*\mathcal{O}_\pi(a(q))\big) \boxtimes \mathcal{O}_\mu(-1) \supseteq \big(\mathcal{S}^\vee \otimes \mathcal{O}_\rho(a(q)-1)\big) \boxtimes \mathcal{O}_\mu(-1)\] giving, respectively, the equations of the \((q;a)\)-tic hypersurfaces parameterized by \(\mathbf{P}\mathcal{M}\), and the corresponding degree \(a(q)\) and \(a(q) - 1\) equations of \(\mathcal{X}_1 \times_S \mathbf{P}\mathcal{M}\) over \(\mathbf{P}\mathcal{M}\). The composition \[\begingroup \def\mathaccent##1##2{ \kern 0.8\dimexpr\macc@kerna \overline{\kern-0.8\dimexpr\macc@kerna\macc@nucleus\kern 0.2\dimexpr\macc@kerna} \kern-0.2\dimexpr\macc@kerna } \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a \macc@nested@a\relax 111{\sigma} \endgroup _1 \colon \mathcal{O}\to \begingroup \def\mathaccent##1##2{ \kern 0.8\dimexpr\macc@kerna \overline{\kern-0.8\dimexpr\macc@kerna\macc@nucleus\kern 0.2\dimexpr\macc@kerna} \kern-0.2\dimexpr\macc@kerna } \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a \macc@nested@a\relax 111{\mathcal{E}} \endgroup _1 \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \mathop{\mathrm{pr}}_1^*\mathcal{E}_1/ (\mathcal{S}^\vee \otimes \mathcal{O}_\rho(a(q)-1)) \boxtimes \mathcal{O}_\mu(-1)\] of \(\mathop{\mathrm{pr}}_1^*\sigma_1\) with the quotient map \(\mathop{\mathrm{pr}}_1^*\mathcal{E}_1 \to \begingroup \def\mathaccent##1##2{ \kern 0.8\dimexpr\macc@kerna \overline{\kern-0.8\dimexpr\macc@kerna\macc@nucleus\kern 0.2\dimexpr\macc@kerna} \kern-0.2\dimexpr\macc@kerna } \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a \macc@nested@a\relax 111{\mathcal{E}} \endgroup _1\) defines the desired component of penultimate tangents.

To keep notation consistent with what follows, assume additionally that there is given an \(r\)-planing \(\mathcal{P} \subseteq \mathcal{X}\) as in [tlines-planing]. Let \[S' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathcal{P} \times_S \mathbf{P}\mathcal{M} \;\;\text{and}\;\; \mathbf{P}\mathcal{V}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{P}\mathcal{T} \times_{\mathcal{X}} S' = \mathbf{P}\mathcal{T}\rvert_{\mathcal{P}} \times_S \mathbf{P}\mathcal{M}\] where the second projection \(\pi' \colon \mathbf{P}\mathcal{V}' \to S'\) exhibits the fibre product as the projective bundle on \(\mathcal{V}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\nu^*(\mathcal{T}\rvert_{\mathcal{P}})\) where \(\nu \colon S' \to \mathcal{P}\) is the structure map. View \(\mathbf{P}\mathcal{V}'\) as a closed subscheme of \(\mathbf{P}\mathcal{T} \times_S \mathbf{P}\mathcal{M}\) and let \(\sigma' \colon \mathcal{O}_{\mathbf{P}\mathcal{V}'} \to \mathcal{E}'\) be the restriction of the section \(\begingroup \def\mathaccent##1##2{ \kern 0.8\dimexpr\macc@kerna \overline{\kern-0.8\dimexpr\macc@kerna\macc@nucleus\kern 0.2\dimexpr\macc@kerna} \kern-0.2\dimexpr\macc@kerna } \macc@depth\@ne \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax} \macc@set@skewchar\relax \let\mathaccentV\macc@nested@a \macc@nested@a\relax 111{\sigma} \endgroup _1\) above. The scheme of penultimate tangents associated with \(\mathcal{X}\) and the subbundle \(\mathrm{S}^a(\mathcal{V}^\vee) \otimes \mathcal{M}\) over \(\mathcal{P}\) is the vanishing locus \[\mathcal{X}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \mathbf{PenTa}(\mathcal{X};\mathrm{S}^a(\mathcal{V}^\vee) \otimes \mathcal{M})\rvert_{\mathcal{P}} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \mathrm{V}(\sigma' \colon \mathcal{O}_{\mathbf{P}\mathcal{V}'} \to \mathcal{E}') \subseteq \mathbf{P}\mathcal{V}'\] of the section \(\sigma'\) in \(\mathbf{P}\mathcal{V}'\). All of this data fits into a commutative diagram of schemes \[\begin{tikzcd} \mathcal{X}_1\rvert_{\mathcal{P}} \times_{\mathcal{P}} S' \rar[symbol={\subseteq}] \dar &[-2em] \mathcal{X}' \rar[symbol={\subseteq}] &[-2em] \mathbf{P}\mathcal{V}' \rar["\pi'"'] \dar["\mathop{\mathrm{pr}}_1"'] & S' \dar["\nu"] \\ \mathcal{X}_1\rvert_{\mathcal{P}} \ar[rr,hook] && \mathbf{P}\mathcal{T}\rvert_{\mathcal{P}} \rar["\rho"] & \mathcal{P} \rar[symbol={\subseteq}] &[-2em] \mathcal{X} \rar[symbol={\subseteq}] &[-2em] \mathbf{P}\mathcal{V} \rar["\pi"] & S\makebox[0pt][l]{\,.} \end{tikzcd}\]

The defining equations \(\sigma'\) of \(\mathcal{X}'\) are essentially a subset of the equations \(\sigma_1\) defining \(\mathcal{X}_1\), whence the containment relation in the top left. This relationship between \(\mathcal{X}'\) and \(\mathcal{X}_1\) further means that the \((q;\mathbf{a}_1)\)-tic structure on \(\mathcal{X}_1\) induces a \((q;\mathbf{a}')\)-tic structure on \(\mathcal{X}'\):

5.4 Induced planings↩︎

— Let \(\mathcal{X}\) be a family of \((q;\mathbf{a})\)-tic schemes in a projective bundle \(\pi \colon \mathbf{P}\mathcal{V} \to S\). An \(r\)-planing \(\mathcal{P} \subseteq \mathcal{X}\) induces a canonical \((r-1)\)-planing of the scheme \(\mathcal{X}_1\rvert_{\mathcal{P}}\) of pointed lines over \(\mathcal{P}\) from [tlines-pointed-lines]: The twisted tangent bundle of \(\mathcal{P}\) provides a rank \(r\) subbundle \[\mathcal{T}_{\mathcal{P}} \otimes \mathcal{O}_\pi(-1) \subseteq \mathcal{T}_{\mathbf{P}\mathcal{V}} \otimes \mathcal{O}_\pi(-1)\rvert_{\mathcal{P}} = \mathcal{T}\rvert_{\mathcal{P}}\] whose associated projective bundle \(\mathcal{P}_1\) is contained in \(\mathcal{X}_1\rvert_{\mathcal{P}}\); geometrically, \(\mathcal{P}_1\) parameterizes pointed lines \((x, \ell)\) where \(\ell \subseteq \mathcal{P}\). Any scheme \(\mathcal{X}'\) of penultimate tangents over \(\mathcal{P}\) as constructed in [tlines-penta-family] also inherits an \((r-1)\)-planing: simply set \(\mathcal{P}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathcal{P}_1 \times_{\mathcal{P}} S'\) and observe that \(\mathcal{X}_1\rvert_{\mathcal{P}} \times_{\mathcal{P}} S' \subseteq \mathcal{X}'\).

5.5 Genericity of the induced families↩︎

— The remainder of this section is concerned with genericity properties of the families \(\mathcal{P}_1 \subseteq \mathcal{X}_1\rvert_{\mathcal{P}}\) of pointed lines over \(\mathcal{P}\), and \(\mathcal{P}' \subseteq \mathcal{X}'\) of penultimate tangents associated with a generic family, in the sense of [families-genericity], of \(r\)-planed \((q;\mathbf{a})\)-tic schemes \(\mathcal{P} \subseteq \mathcal{X}\) over an integral base \(S\). First, to see that the families \(\mathcal{P}_1 \subseteq \mathcal{X}_1\rvert_{\mathcal{P}}\) and \(\mathcal{P}' \subseteq \mathcal{X}'\) are themselves generic, consider the tautological situation over the incidence correspondence of \(r\)-planes and \((q;\mathbf{a})\)-tic schemes:

For an analogous statement for penultimate tangents, observe that the \((q;\mathbf{a})\)-tic tensor \(\boldsymbol{\alpha} \colon \mathcal{O}_S \to \mathcal{A}\) defining the tautological family \(\mathcal{X} \subseteq \mathbf{P}^n_S\) takes values in a split bundle of the form \[\mathcal{A} = \bigoplus\nolimits_{a \in \mathbf{a}} \mathrm{S}^a(\mathcal{V}^\vee)\] where \(\mathcal{V}\) itself is a trivial \(\mathcal{O}_S\)-module of rank \(n+1\). Choose a summand \(\mathrm{S}^a(\mathcal{V}^\vee) \subseteq \mathcal{A}\) with nonlinear and reduced profile \(a\), apply the constructions of [tlines-penta-family] and [tlines-induced-planing], and let \(\mathcal{P}' \subseteq \mathcal{X}'\) be the resulting family of \((r-1)\)-planed \((q;\mathbf{a}')\)-tic schemes, with \(\mathbf{a}' = \mathbf{a}_1 \setminus (a,a-1)\), over \(S' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathcal{P}\).

Consider an arbitrary family \(\mathcal{P} \subseteq \mathcal{X}\) of \(r\)-planed \((q;\mathbf{a})\)-tic schemes over an integral base \(S\). If the family \(\mathcal{P} \subseteq \mathcal{X}\) is generic in the sense of [families-genericity], then so too are the families \(\mathcal{P}_1 \subseteq \mathcal{X}_1\rvert_{\mathcal{P}}\) of pointed lines over \(\mathcal{P}\) with its structure from 1, and—whenever \(\mathbf{a}_{\mathrm{nlr}} \neq \varnothing\)—any family \(\mathcal{P}' \subseteq \mathcal{X}'\) of penultimate tangents as constructed in [tlines-penta-family] and 1:

Combined with numerical assumptions on the ambient projective bundle dimension \(n\) and the integer \(r\), genericity of the family \(\mathcal{P}' \subseteq \mathcal{X}'\) yields geometric genericity properties of the families \(\mathcal{X}_1\rvert_{\mathcal{P}} \to \mathcal{P}\) and \(\mathcal{X}' \to S\). The most useful for what follows is that whenever \(n\) is sufficiently large depending on \(r\) and \(\mathbf{a}\), the general fibre of either family has its expected dimension:

6 Residual point map↩︎

As mentioned at the beginning of §5, extracting the residual point of intersection provides a rational map from a scheme of penultimate lines back to the original projective scheme. The aim of this section is to construct this rational map in the relative setting, for the family \(\mathcal{X}'\) of penultimate lines constructed in [tlines-penta-family], and to study when the resulting map \(\mathop{\mathrm{res}}\colon \mathcal{X}' \dashrightarrow \mathcal{X}\) is dominant.

To begin, let \(\mathcal{P} \subseteq \mathcal{X}\) be a family of \(r\)-planed \((q;\mathbf{a})\)-tic schemes in a projective bundle \(\pi \colon \mathbf{P}\mathcal{V} \to S\). Assume that \(\mathcal{A}_{\mathrm{nlr}} \neq 0\), choose a subbundle \(\mathrm{S}^a(\mathcal{V}^\vee) \otimes \mathcal{M} \subseteq \mathcal{A}_{\mathrm{nlr}}\) as in 1, and let \(\mathcal{X}' \to S'\) be the associated family of penultimate tangents as in [tlines-penta-family]. To give an explicit description of the residual point map, represent geometric points of \(\mathcal{X}'\) as quintuples \((s, x, \ell, H, Y)\) where

• \(s\) is a geometric point of \(S\);

• \(x \in \ell\) is a pointed line in \(\mathbf{P}\mathcal{V}_s\) with \(x \in \mathcal{P}_s\);

• \(H\) is a \((q;a)\)-tic hypersurface parameterized by the linear system \(\mathbf{P}\mathcal{M}_s\) as in 1; and

• \(Y\) is the vanishing locus of the remaining equations so that \(\mathcal{X}_s = Y \cap H\) is a presentation of \(\mathcal{X}_s\) as a \((q;\mathbf{a})\)-tic scheme.

This data is subject to the conditions that \(\ell \subseteq Y\) and \[\mathop{\mathrm{mult}}_x(\ell \cap H) \geq a(q) - 1.\] Note \(H\) is not uniquely determined in this representation. However, any other \(H'\) representing the same point of \(\mathcal{X}'\) has equation differing from that of \(H\) by a \((q;a)\)-tic polynomial in the ideal of \(Y\). In particular, since \(\ell \subseteq Y\), the multiplicity condition is well-posed. The residual point map is now described, and constructed, as follows:

When the families \(\mathcal{X}_1\rvert_{\mathcal{P}} \to \mathcal{P}\) and \(\mathcal{X}' \to S'\) of pointed lines and penultimate tangents associated with an \(r\)-planed filtered family \(\mathcal{P} \subseteq \mathcal{X}\) of \((q;\mathbf{a})\)-tic complete intersections are themselves generically complete intersections, 1 provides a rational map \(\mathop{\mathrm{res}}\colon \mathcal{X}' \dashrightarrow \mathcal{X}\) which takes a penultimate tangent to its residual point of intersection. The next goal is to show that \(\mathop{\mathrm{res}}\) is dominant whenever \(r\) is sufficiently large depending only on \(\mathbf{a}\). In the following statement, a property is said to hold fibrewise in a family over \(S\) if the property holds upon restriction to each closed point of \(S\).

Begin with a few reductions: If \(\mathbf{a}_{\mathrm{nlr}} = \varnothing\), then the statement is empty and there is nothing to prove. So assume otherwise and fix any family \(\mathcal{X}'\) of penultimate tangents as constructed in [tlines-penta-family]. Using the fact that the first projection \(\mathcal{X} \times_S \mathbf{P}\mathcal{M} \to \mathcal{X}\) is surjective, \(S\) may be replaced by the linear system \(\mathbf{P}\mathcal{M}\) of hypersurfaces relative to which \(\mathcal{X}'\) is constructed so as to assume \(\mathcal{M} = \mathcal{O}_S\) and \(\mathcal{X}'\) is a family over \(S' = \mathcal{P}\). Since, fibrewise over \(S\), \(\mathcal{X}'\) is obtained from \(\mathcal{X}_1\) by omitting two relatively ample divisors over their common base \(\mathcal{P}\), the hypothesis that \(\mathcal{X}_1\) is of expected dimension implies that \(\mathcal{X}'\) is also of expected dimension. This guarantees, via 1, that the residual point map \(\mathop{\mathrm{res}}\colon \mathcal{X}' \dashrightarrow \mathcal{X}\) exists, and even that its indeterminacy locus does not contain any fibre over \(S\). Given this, since the statement is fibrewise over \(S\), it suffices to consider the absolute case over \(\mathop{\mathrm{Spec}}\mathbf{k}\).

For the remainder of the proof, then, let \(P \subseteq X\) be an \(r\)-planed \((q;\mathbf{a})\)-tic complete intersection in \(\mathbf{P}^n\) over \(\mathbf{k}\). Write \(\mathbf{a} = (a_1,\ldots,a_c)\) and fix a presentation of the form \[X = H_1 \cap \cdots \cap H_c \;\;\text{where}\; H_i\;\text{is a \((q;a_i)\)-tic hypersurface}.\] Perhaps after reordering, assume that \(a_1 \in \mathbf{Prfl}\) is nonlinear and reduced, and let \(X'\) be the scheme of penultimate tangents with respect to \(H_1\).

For each point \(y \in X \setminus P\), consider the closed subscheme of \(P\) given by \[Z_y \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \{z \in P : \mathop{\mathrm{mult}}_z(\ell_{y,z} \cap H_1) \geq a_1(q) - 1\;\text{and}\; \ell_{y,z} \subset H_i \;\text{for}\; 2 \leq i \leq c \}\] where \(\ell_{y,z}\) is the line between \(y\) and \(z\). The task is to show that the open subscheme \(Z_y^\circ \subseteq Z_y\) parameterizing lines intersecting \(H_1\) at \(z\) with multiplicity exactly \(a_1(q) - 1\) is nonempty for general \(y\). Toward this, observe that:

Equations for \(Z_y\) are simple to describe, and yield the following dimension estimate:

7 Unirationality↩︎

The goal of this section is to establish Theorem 2, showing that general \((q;\mathbf{a})\)-tic complete intersection is unirational once its dimension is sufficiently large, depending only on the multi-profile \(\mathbf{a}\). The proof proceeds by inductively simplifying the equations of the tautological family of \((q;\mathbf{a})\)-tic complete intersections via the constructions of §§4–6. Induction takes place over the set \(\Pi\) of all multi-profiles equipped with a somewhat complicated partial ordering; the ordering is designed to keep track of the multi-profiles that appear after passing to one of the following three constructions: Frobenius descent as in 1; removing linear equations as in 1; and passing to the family of penultimate tangents as in 1.

7.1 Ordering multi-profiles↩︎

— Let \(\Pi\) be the set of all multi-profiles, and consider the relation \(\preceq^\Pi\) defined as follows: The cover relations \(\mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\) for this relation come in three flavours, depending on which parts of the canonical type decomposition from 1 are present:

1. If \(\mathbf{a} = \mathbf{a}_{\mathrm{pow}}\), then set \(\mathbf{a}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{a}/t \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =(a/t : a \in \mathbf{a})\).

2. If \(\mathbf{a} = \mathbf{a}_{\mathrm{lin}} \sqcup \mathbf{a}_{\mathrm{pow}}\) and \(\mathbf{a}_{\mathrm{lin}} \neq \varnothing\), then set \(\mathbf{a}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\mathbf{a} \setminus \mathbf{a}_{\mathrm{lin}}\).

3. If \(\mathbf{a}_{\mathrm{nlr}} \neq \varnothing\), then let \(a_0 \in \mathbf{a}_{\mathrm{nlr}}\) be any element with maximal coefficient sum \(a_0(1)\), and set \[\mathbf{a}' \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = (b \in \mathbf{Prfl}: 0 \prec b \preceq a \;\;\text{for}\; a \in \mathbf{a}) \setminus (a_0,a_0 -1).\]

In general, two multi-profiles satisfy \(\mathbf{a}' \preceq^\Pi \mathbf{a}\) if and only if \(\mathbf{a}' = \mathbf{a}\) or else they are connected by a finite sequence of the above cover relations: \[\mathbf{a}' = \mathbf{a}_n \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}_{n-1} \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \cdots \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}_1 \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}_0 = \mathbf{a}.\]

7.2 Examples↩︎

— The following are a few examples illustrating properties of the poset \((\Pi,\preceq^\Pi)\):

1. If \(a(t) = d\) is a constant, then the interval \([\varnothing, d]^\Pi\) between \(\varnothing\) and \(d\) is totally ordered. Explicitly, the Hasse diagrams for \(3 \leq d \leq 5\) take the form: \[(3) - (1) - \varnothing, \quad (4) - (2, 1) - (1) - \varnothing, \quad (5) - (3, 2, 1) - (2, 1,1,1) - (1,1,1) - \varnothing.\] To depict the Hasse diagram for \(d = 6\), write \(k^m\) for the profile \(k\) appearing \(m\) times: \[(6) - (4,3,2,1) - (3,2^3,1^4) - (2^3,1^8) - (2^2,1^{10}) - (2,1^{11}) - (1^{11}) - \varnothing.\]

2. The Hasse diagram for the interval \([\varnothing, 1+t]^\Pi\) is: \((1+t) - (1) - \varnothing\).

3. The cover relation [unirational-poset][unirational-poset46pow] appears in the Hasse diagram for \([\varnothing, 1+2t]^\Pi\): \[(1+2t) - (1+t, t, 1) - (t,1,1) - (t) - (1) - \varnothing.\]

4. Consider the multi-profile \(\mathbf{a} \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =((q+1)t + 1, t^2+(q+1))\) of \(\sqsubseteq\)-incomparable profiles of numerical degree \(q^2+q+1\) from [qatics-ordering-properties][qatics-ordering-properties46not-total]. Specializing to \(q = 2\), the two multi-profiles covered by \(\mathbf{a}\) via the relation [unirational-poset][unirational-poset46nlr] are obtained from \[\mathbf{a}_1 = (3t+1, 3t, 2t+1, 2t, t+1, t, 1) \cup (t^2+3, t^2+2, t^2+1, t^2, 3, 2, 1)\] by omitting either \((3t+1, 3t)\) or else \((t^2+3, t^2+2)\).

5. By considering a slight variant of [unirational-poset-examples46incomparable], it is possible to see that lengths of paths from \(\mathbf{a}\) to \(\varnothing\) need not be the same. For instance, this can be seen with the pair \(\mathbf{a} = ((q+2)t + 1, t^2 + (2q+1))\) of \(\sqsubseteq\)-incomparable profiles of numerical degree \(q^2+2q+1\).

6. Maximal coefficient sums in a multi-profile does not necessarily drop along a cover relation of the form [unirational-poset][unirational-poset46nlr]. This is because the maximum may be achieved by a member of \(\mathbf{a}_{\mathrm{pow}}\). For example, consider the Hasse diagram for \(\mathbf{a} = (2,3t)\): \[(2,3t) - (3t) - (3) - (1) - \varnothing.\]

7.3 An invariant↩︎

— Basic properties of \((\Pi,\preceq^\Pi)\) require some effort to establish since, for example, multi-profiles may grow bigger along the cover relations defined in [unirational-poset] and maximal coefficient sums of profiles does not necessarily decrease along \(\preceq^\Pi\) as in [unirational-poset-examples][unirational-poset-examples46degree-drop]. To address these difficulties, consider the three integer-valued functions \[\delta(\mathbf{a}) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\max\{\deg_t a(t) : a \in \mathbf{a}\}, \;\; \sigma(\mathbf{a}) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\max\{a(1) : a \in \mathbf{a}_{\mathrm{nlr}}\}, \;\; \mu(\mathbf{a}) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\#\{a \in \mathbf{a}_{\mathrm{nlr}} : a(1) = \mu_{\mathrm{nlr}}(\mathbf{a})\},\] where the maximum of an empty set is \(0\). Define \(\phi \colon \Pi \to \mathbf{Z}_{\geq 0}^4\) by \(\phi(\mathbf{a}) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =(\delta(\mathbf{a}), \sigma(\mathbf{a}), \mu(\mathbf{a}), \#\mathbf{a}_{\mathrm{nlr}})\). A case analysis shows that \[\text{if \(\mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\) is a cover relation in \(\Pi\), then \(\phi(\mathbf{a}') <^{\mathrm{lex}} \phi(\mathbf{a})\)},\] where \(<^{\mathrm{lex}}\) is the lexicographical ordering on \(\mathbf{Z}_{\geq 0}^4\). Thus this invariant provides a relation preserving function \(\phi \colon (\Pi, \preceq^\Pi) \to (\mathbf{Z}_{\geq 0}^4, \leq^{\mathrm{lex}})\). This makes it possible to establish the basic properties of \(\preceq^\Pi\):

7.4 Numbers↩︎

— Inductively define numerical functions on the poset \((\Pi, \preceq^\Pi)\) as follows: Let \(r \geq 0\) be an integer and \(\mathbf{a}\) be a multi-profile. Assuming that \(\mathbf{a}_{\mathrm{nlr}} \neq \varnothing\), define \[\begin{align} r(\mathbf{a}) & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \max\{r_0(\mathbf{a})\} \cup \{r(\mathbf{a}') + 1 : \mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a} \}, \\ n_1(\mathbf{a},r) & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \max\{2r-1+ \#\mathbf{a}_1\} \cup \{n_1(\mathbf{a}', r-1) + 1 : \mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\}, \;\text{and} \\ n_2(\mathbf{a},r) & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \max \Big\{ \big\lceil r+ \frac{1}{r} \sum\nolimits_{a \in \mathbf{a}} \prod\nolimits_{j \geq 0} \binom{a_j + r}{r} -\frac{1}{r}\#\mathbf{a}\big\rceil\Big\} \cup \{n_2(\mathbf{a}', r-1) + 1 : \mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\}, \end{align}\] where \(r_0(\mathbf{a})\) is defined in 1, \(\mathbf{a}'\) in the second set ranges over elements of \(\Pi\) covered by \(\mathbf{a}\) as in [unirational-poset], and \(\lceil \cdot \rceil\) denotes the ceiling function. One may verify that \(n_1(\mathbf{a},r) \leq n_2(\mathbf{a},r)\) unless \(\mathbf{a} = (2t^k) \cup \mathbf{a}'\) where \(\mathbf{a}'\) consist of profiles of the form \(t^m\). In the case that \(\mathbf{a}_{\mathrm{nlr}} = \varnothing\), define \[\begin{align} r(\mathbf{a}) & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \max\{r_0(\mathbf{a})\} \cup \{r(\mathbf{a}') : \mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\}, \\ n_1(\mathbf{a},r) & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \max\{2r-1+ \#\mathbf{a}_1\} \cup \{n_1(\mathbf{a}', r) : \mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\},\;\text{and} \\ n_2(\mathbf{a},r) & \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} = \max \Big\{ \big\lceil r+ \frac{1}{r} \sum\nolimits_{a \in \mathbf{a}} \prod\nolimits_{j \geq 0} \binom{a_j + r}{r} -\frac{1}{r}\#\mathbf{a}\big\rceil\Big\} \cup \{n_2(\mathbf{a}', r) : \mathbf{a}' \prec\mathrel{\mkern-5mu}\mathrel{\cdot}^\Pi \mathbf{a}\}, \end{align}\] where the notation is as before. Here, there is precisely one multi-profile \(\mathbf{a}'\) covered by \(\mathbf{a}\), and a direct computation shows that \(r_0(\mathbf{a}) = r_0(\mathbf{a}')\), and hence \(r(\mathbf{a}) = r(\mathbf{a}')\), and also \(n_i(\mathbf{a}',r) = n_i(\mathbf{a},r) - \#\mathbf{a}_{\mathrm{lin}}\) for \(i = 1,2\). Finally, for all \(\mathbf{a} \in \Pi\) and \(r \geq 0\), define \[n_0(\mathbf{a},r) \mathrel{\rlap{ \raisebox{0.3ex}{\m@th\cdot}} \raisebox{-0.3ex}{\m@th\cdot}} =\max\{n_1(\mathbf{a},r), n_2(\mathbf{a},r)\}\]

With these, it is possible to formulate the precise unirationality result:

7.5 Example↩︎

— The bound \(n_0(\mathbf{a})\) may be computed for small multi-profiles \(\mathbf{a}\). Some examples: \[n_0(t+1) = 4, \quad n_0(2,t+1) = 9, \quad n_0(t+1,t+1) = 13, \quad n_0(t^2 + t + 1) = 48.\] The bounds obtained for constant profiles \(a(t) = d\), corresponding to degree \(d\) hypersurfaces as in [qatics-examples][qatics-examples46constant], are surprisingly small: \[\begin{gather} n_0(3) = 4, \quad n_0(4) = 9, \quad n_0(5) = 22, \quad n_0(6) = 160, \quad n_0(7) = 20376, \\ n_0(8) = 11914188890, \quad n_0(9) = 8616199237736295920955120, \end{gather}\] and \(n_0(10) = 192884152577980851363553858004926940342106493833715693762179 \approx 10^{59}\). In comparison, the best bounds \(n'_0(d)\) for the classical unirationality construction are from [20], and they give \[\begin{gather} n'_0(3) = 3, \quad n'_0(4) = 20, \quad n'_0(5) = 8855, \quad n'_0(6) = 454205040715033146, \\ n'_0(7) \approx 10^{103}, \quad n'_0(8) \approx 10^{717}, \quad n'_0(9) \approx 10^{5738}, \quad n'_0(10) \approx 10^{51641}. \end{gather}\] In the companion paper [23], it is shown that the integer \(n_0(d)\) is on the order of \(2^{d2^d}\), growing significantly slower than \(n_0'(d) \sim 2^{d!}\).

References↩︎