1 Introduction↩︎

Throughout this article we work over the field \(\mathbb{C}\) of complex numbers, a variety and a curve are irreducible and reduced, and \(\mathbb{P}V\) denotes the projective space of \(1\)-dimensional subspaces of \(V^\ast\), hence \(V\) is the space of linear forms on \(\mathbb{P}V\). Let \(X\subseteq\mathbb{P}V\) be a projective variety that is nondegenerate, which means that its linear span is \(\langle X\rangle=\mathbb{P}V\). Write \(I(X)=\bigoplus_{j\geq 2}I(X)_j\) for the ideal of \(X\subseteq\mathbb{P}V\).

Our focus is on the tangent variety \[\tau X=\overline{\bigcup_{z\in\operatorname{Sm}X}\mathbb{T}_zX}\subseteq\mathbb{P}V\] to \(X\subseteq\mathbb{P}V\) and its quadratic equations, where \(\operatorname{Sm}X\) is the smooth locus of \(X\), and \(\mathbb{T}_zX\) is the projective tangent space to \(X\) at \(z\in\operatorname{Sm}X\). More generally, we consider the \(q\)-secant variety \(\sigma_q\tau^kX\) to the \(\emph{k-osculating variety}\) \(\tau^kX\) to \(X\subseteq\mathbb{P}V\) for arbitrary integers \(q\geq 1\) and \(k\geq 0\). Our treatment applies not only to \(\tau X\) but also to \(\sigma_q\tau^kX\), and in fact, taking \(\sigma_q\tau^kX\) into account is technically required. See 2.2 for the definition of \(\sigma_q\tau^kX\), and note that \(\sigma_1\tau^1X=\tau X\).

Suppose that we are given a linear map of the form \[T:V_1\otimes\cdots\otimes V_{2k+2}\to V,\] namely a \((2k+2)\)-way tensor of linear forms, and associate to it the following composition map \(b_{q+1}(T)\). \[\begin{tikzcd} {\textstyle\bigwedge}^{q+1}V_1\otimes\cdots\otimes{\textstyle\bigwedge}^{q+1}V_{2k+2} \ar[rr,"b_{q+1}(T)"] \ar[dr,"\mathrm{canonical}",hook,swap] & & S^{q+1}V \\ & S^{q+1}(V_1\otimes\cdots\otimes V_{2k+2}) \ar[ur,"T",swap] \end{tikzcd}\] For example, if \(k=0\), then \(T\) corresponds to a matrix \(M\) with entries in \(V\), namely a matrix of linear forms, and the image of \(b_{q+1}(M)\) is spanned by the \((q+1)\)-minors of \(M\).

Now our first main theorem presents a systematic construction of quadratic equations for tangent varieties via \(4\)-way tensors of linear forms. It naturally generalizes the well-known technique [1] that yields determinantal equations for projective varieties and higher secant varieties using minors of matrices of linear forms.

For the definition and structure of \(X\)-multiplicative tensors of linear forms, consult with 26 and 29. As a central example, if a very ample line bundle \(L\) on \(X\) is factored as \(L=L_1\otimes\cdots\otimes L_{2k+2}\) in the Picard group \(\operatorname{Pic}(X)\), then for the complete embedding \(X\subseteq\mathbb{P}H^0(L)\) the multiplication map \[H^0(L_1)\otimes\cdots\otimes H^0(L_{2k+2})\to H^0(L)\] is an \(X\)-multiplicative \((2k+2)\)-way tensor of linear forms. For a tensor of this type 1 works as follows.

We point out that 1 can provide enough quadratic equations to span \(I(\tau C)_2\) for an arbitrary completely embedded smooth projective curve \(C\subset\mathbb{P}H^0(L)\) when \(L\) has sufficiently large degree. Recall the fact [1] that if \(C\subset\mathbb{P}H^0(L)\) has genus \(g\) and degree \(\geq 4g+2\), then \(I(C)\) is generated by the \(2\)-minors of some matrix of linear forms, together with Eisenbud-Koh-Stillman’s problem [1] below.

It has been solved set-theoretically [2] and scheme-theoretically [3]. In fact, its ideal-theoretic validity is already well known to experts; thus, the main challenge lies in finding an effective bound.

The following can be considered as an affirmative answer to an analogue of the conjecture for tangent varieties \(\tau C\subset\mathbb{P}H^0(L)\).

In the case \(k=0\) it resolves the Eisenbud-Koh-Stillman conjecture¹ since the minimal generators of \(I(\sigma_qC)\) have degree \(q+1\) whenever \(L\) has degree \(\geq 2g+2q\) by [5], and its lower bound condition \(\deg L\geq(2k+2)(2g+q)\) is very reasonable along the line of Eisenbud-Koh-Stillman’s one \(\deg L\geq 2(2g+1)\) for the case \((q,k)=(1,0)\).²

Furthermore, the quadratic equations of tangent varieties in 1 lead us to an unexpected byproduct about the nondefectiveness of \(\tau X\) and \(\sigma_2X\), based on Fulton-Hansen’s result [6] (cf.[7]). Compare it with the fact that the conclusion below holds if the embedding line bundle \(L\) is \(3\)-very ample, meaning that it separates every finite subscheme of length \(4\) in \(X\).

It can be rephrased as follows: If \(L\) is more positive than a product of four pencils, then \(\tau X\) and \(\sigma_2X\) have the expected dimensions.

The next main theorem tells us that the quadratic equations of \(\tau X\subseteq\mathbb{P}V\) above carry rich linear syzygies. Consider the minimal free resolution \[\begin{tikzcd} F_0 & \ar[l] F_1 & \ar[l] \cdots & \ar[l] F_i=\displaystyle{\bigoplus_{j\in\mathbb{Z}}}K_{i,j}(I(\tau X))\otimes_{\mathbb{C}}S(-i-j) & \ar[l] \cdots \end{tikzcd}\] of \(I(\tau X)\) over the symmetric algebra \(S:=S^\ast V\), where the vector spaces \(K_{i,j}(I(\tau X))\), called the Koszul cohomology groups of \(I(\tau X)\), encode information about bases of the free modules \(F_i\). Linear syzygies of \(\tau X\subseteq\mathbb{P}V\) are, roughly speaking, elements of the \(K_{p,2}(I(\tau X))\) and explain the interaction between the quadratic equations of \(\tau X\subseteq\mathbb{P}V\): \[K_{p,2}(I(\tau X))=\ker({\textstyle\bigwedge}^pV\otimes I(\tau X)_2\to{\textstyle\bigwedge}^{p-1}V\otimes I(\tau X)_3)\] for a Koszul type differential.

To describe the aforementioned linear syzygies, recall Schur modules \(S^\lambda U\), of a vector space \(U\), indexed by partitions \(\lambda=(\lambda_1\geq\lambda_2\geq\cdots)\vdash t\) of integers \(t\geq 0\), where if \(\lambda=(t)\) (resp.\(\lambda=(1,\ldots,1)\vdash t\)), then we obtain the \(t\)-th symmetric power \(S^tU\) (resp.the \(t\)-th exterior power \({\textstyle\bigwedge}^tU\)). Unless \(\lambda_{\dim U}=0\), the Schur module \(S^\lambda U\) forms an irreducible finite-dimensional representation of the general linear group \(\operatorname{GL}(U)\). Let \(p\geq 0\) be an integer and \(p_\ast=(p_1,\ldots,p_{2k+2})\vdash p\) be a \((2k+2)\)-tuple of nonnegative integers that sum up to \(p\): \[p_1+\cdots+p_{2k+2}=p,\] namely an ordered partition of \(p\). Now we assign to \(p_\ast\) the tensor product \[B_{p_\ast,q+1}(V_1,\ldots,V_{2k+2})=S^{\lambda^1}V_1\otimes\cdots\otimes S^{\lambda^{2k+2}}V_{2k+2},\quad\lambda^i:=(p_i+1,1,\ldots,1)\vdash p+q+1,\] of Schur modules of vector spaces \(V_1,\ldots,V_{2k+2}\).

Note that for every integer \(j<q+1\) the component \(I(\sigma_q\tau^kX)_j\) always vanishes, hence \(K_{i,j}(I(\sigma_q\tau^kX))=0\) for all \(i\geq0\), and observe that Condition ?? is nothing but the nonvanishing of \(B_{p_\ast,q+1}(V_1,\ldots,V_{2k+2})\).

The naturality of \(b_{p_\ast,q+1}(T)\) can be demonstrated by the case of Segre varieties of the form \[\mathbb{P}V_1\times\cdots\times\mathbb{P}V_{2k+2}\subset\mathbb{P}(V_1\otimes\cdots\otimes V_{2k+2}),\] including the sufficiency.

The foregoing results collectively constitute a new matryoshka structure. Here, a matryoshka structure means a sequence of parallel facts indexed by integers \(q\geq 1\) such that the \(q\)-th one holds for \(q\)-secant varieties. It is very interesting that higher secant varieties exhibit various matryoshka structures. We refer to [5], [8], [9], [10], and [11] for theorems and [12] for a conjecture.

One of the motivations for this study is Aprodu-Farkas-Papadima-Raicu-Weyman’s proof of Green’s conjecture for general canonical curves. For any smooth canonical curve \(C\subset\mathbb{P}H^0(\omega_C)\) of degree \(g\geq 3\), Green suggested a conjectural criterion for the vanishing/nonvanishing of \(K_{i,j}(I(C))\) in terms of the Clifford index of \(C\).

Its validity for general curves was proved by Voisin ([14] and [15]) and by Aprodu-Farkas-Papadima-Raicu-Weyman ([16]). The method of [16] employs a general hyperplane section \[\tau(\nu_g(\mathbb{P}^1))\cap H\subset H=\mathbb{P}^{g-1}\] of the tangent variety to the rational normal curve \(\nu_g(\mathbb{P}^1)\subset\mathbb{P}^g\) of degree \(g\). By relying on a semicontinuity argument applied to the moduli space of pseudostable curves of arithmetic genus \(g\), a detailed analysis of the linear syzygies of \(\tau(\nu_d(\mathbb{P}^1))\subset\mathbb{P}^g\) completes the proof.

However, the equations of tangent varieties have been surprisingly unexplored in the literature except certain special cases. Settling a conjecture [17] of Landsberg and Weyman, Oeding and Raicu [18] completely described the equations of tangent varieties to Segre-Veronese varieties, and in particular, they showed that for every \(4\)-factor Segre variety \(Y:=\mathbb{P}V_1\times\cdots\times\mathbb{P}V_4\subseteq\mathbb{P}(V_1\otimes\cdots\otimes V_4)\), the quadrics passing through \(\tau Y\) form the space \[{\textstyle\bigwedge}^2V_1\otimes\cdots\otimes{\textstyle\bigwedge}^2V_4\subset S^2(V_1\otimes\cdots\otimes V_4).\] For other projective varieties, only case-by-case results (such as [17]) are known for tangent varieties, in contrast to higher secant varieties which admit various general frameworks for finding equations, including the determinantal approach.

Another motivation arises from Green-Lazarsfeld classes. For a complete embedding \(X\subseteq\mathbb{P}H^0(L)\) let \(M:H^0(L_1)\otimes H^0(L_2)\to H^0(L)\) be a multiplication matrix of complete linear systems on \(X\) with \(h^0(L_1)\geq 2\) and \(h^0(L_2)\geq 2\). Then the \(2\)-minors of \(M\) vanish on \(X\subseteq\mathbb{P}H^0(L)\), which means that the induced map \[{\textstyle\bigwedge}^2H^0(L_1)\otimes{\textstyle\bigwedge}^2H^0(L_2)\to S^2H^0(L)\] factors through \(I(X)_2\). The determinantal quadrics allow linear syzygies of \(X\subset\mathbb{P}H^0(L)\) to go far away, yielding the nonvanishing \[K_{r_1+r_2-2,2}(I(X))\neq 0,\quad r_i:=\dim|L_i|\geq 1,\] from an explicit nonzero element, namely a Green-Lazarsfeld class [13]. (See [19], [20], and [15] for alternative accesses.) Such a construction also works well for higher secant varieties: \[K_{r_1+r_2-2q,q+1}(I(\sigma_qX))\neq0\] whenever \(r_1\geq q\) and \(r_2\geq q\). The nonvanishing for \(I(X)\) is in fact the origin of Green’s conjecture and a former conjecture called the gonality conjecture which play a central role in the theory of syzygies, and the one for \(I(\sigma_qX)\) generalizes the gonality conjecture to higher secant varieties in terms of the gonality sequence [11].

2 Preliminaries↩︎

In this section we collect useful notions and facts.

2.1 Multilinear algebra↩︎

Let \(U\) be a vector space of finite dimension. Fix a basis \(u=(u_0,u_1,\ldots)\) of \(U\), and take its dual basis \[\partial_u=(\partial_u^0,\partial_u^1,\ldots)\quad\text{or}\quad\partial^u=(\partial^u_0,\partial^u_1,\ldots)\] for \(U^\ast\), where the former (resp.the latter) applies when dealing with \(S^\ast U\) (resp.\({\textstyle\bigwedge}^\ast U\)). For an element \(\alpha=(\alpha_0,\alpha_1,\ldots)\vdash a\) in \(\mathbb{N}^{\dim U}\) with \(\alpha_0+\alpha_1+\cdots=a\),

Similarly, for a subset \(\alpha=\{\alpha_0<\alpha_1<\cdots\}\) of cardinality \(a\) in \(\{0,\ldots,\dim U-1\}\),

Just as \(S^aU^\ast\) forms the space of partial differentials of order \(a\) on \(S^\ast U\), so does \({\textstyle\bigwedge}^aU^\ast\) on \({\textstyle\bigwedge}^\ast U\) in the following way. For any elements \(f\in{\textstyle\bigwedge}^bU\) and \(g\in{\textstyle\bigwedge}^\ast U\) we have \[\partial^u_i(f\wedge g)=(\partial^u_if)\wedge g+(-1)^bf\wedge(\partial^u_ig),\] and the usual product on \({\textstyle\bigwedge}^\ast U^\ast\) corresponds to the composition in reverse order, that is, \[\partial^u_\alpha\wedge\partial^u_\beta=\partial^u_\beta\circ\partial^u_\alpha\] for all \(\alpha,\beta\subseteq\{0,\ldots,\dim U-1\}\).

Consider the coproduct maps \[\Delta:S^{a+b}U\to S^aU\otimes S^bU\quad\text{and}\quad\Delta:{\textstyle\bigwedge}^{a+b}U\to{\textstyle\bigwedge}^aU\otimes{\textstyle\bigwedge}^bU\] which are computed by the formula \[\binom{a+b}{a}\Delta(f)= \begin{cases} \displaystyle{\sum_{\alpha\vdash a}\frac{1}{\alpha!}u^\alpha\otimes\partial_u^\alpha f} & \mathrm{if }f\in S^{a+b}U \\ \displaystyle{\sum_{|\alpha|=a}u_\alpha\otimes\partial^u_\alpha f} & \mathrm{if }f\in{\textstyle\bigwedge}^{a+b}U, \end{cases}\] where either \(\alpha\in\mathbb{N}^{\dim U}\), or \(\alpha\subseteq\{0,\ldots,\dim U-1\}\). They are sections of the product maps \(S^aU\otimes S^bU\to S^{a+b}U\) and \({\textstyle\bigwedge}^aU\otimes{\textstyle\bigwedge}^bU\to{\textstyle\bigwedge}^{a+b}U\), respectively, which amounts to Euler’s homogeneous function theorem.

Koszul differentials are linear transformations \[\delta_{a,b}:{\textstyle\bigwedge}^aU\otimes S^bU\to{\textstyle\bigwedge}^{a-1}U\otimes S^{b+1}U\quad\text{and}\quad \delta^{a,b}:S^aU\otimes{\textstyle\bigwedge}^bU\to S^{a-1}U\otimes{\textstyle\bigwedge}^{b+1}U\] defined as the compositions of \[\begin{array}{cl} \begin{tikzcd} {\textstyle\bigwedge}^aU\otimes S^bU \ar[r,"\Delta\otimes\operatorname{id}"] & {\textstyle\bigwedge}^{a-1}U\otimes U\otimes S^bU \ar[r,"\operatorname{id}\otimes\cdot"] & {\textstyle\bigwedge}^{a-1}U\otimes S^{b+1}U \end{tikzcd} & \text{and} \\ \begin{tikzcd} S^aU\otimes{\textstyle\bigwedge}^bU \ar[r,"\Delta\otimes\operatorname{id}"] & S^{a-1}U\otimes U\otimes{\textstyle\bigwedge}^bU \ar[r,"\operatorname{id}\otimes\wedge"] & S^{a-1}U\otimes{\textstyle\bigwedge}^{b+1}U, \end{tikzcd} \end{array}\] respectively. They constitute chain complexes \[\begin{array}{cl} \begin{tikzcd} \cdots \ar[r] & {\textstyle\bigwedge}^{a+1}U\otimes S^{b-1}U \ar[r,"\delta_{a+1,b-1}"] & {\textstyle\bigwedge}^aU\otimes S^bU \ar[r,"\delta_{a,b}"] & {\textstyle\bigwedge}^{a-1}U\otimes S^{b+1}U \ar[r] & \cdots \end{tikzcd} & \text{and} \\ \begin{tikzcd} \cdots \ar[r] & S^{a+1}U\otimes{\textstyle\bigwedge}^{b-1}U \ar[r,"\delta^{a+1,b-1}"] & S^aU\otimes{\textstyle\bigwedge}^bU \ar[r,"\delta^{a,b}"] & S^{a-1}U\otimes{\textstyle\bigwedge}^{b+1}U \ar[r] & \cdots \end{tikzcd} \end{array}\] that are exact everywhere except at \({\textstyle\bigwedge}^0U\otimes S^0U\) and \(S^0U\otimes{\textstyle\bigwedge}^0U\), respectively, at which the (co)homology groups have dimension one. Write \[Z_{a,b}(U)=\ker\delta_{a,b}\quad\text{and}\quad Z^{a,b}(U)=\ker\delta^{a,b}\] for the groups of cycles.

Schur modules \(S^\lambda U\) are finite-dimensional representations of the general linear group \(\operatorname{GL}(U)\), depending on partitions \(\lambda=(\lambda_1\geq\lambda_2\geq\cdots)\), where \(S^\lambda U=S^tU\) if \(\lambda=(t)\), and \(S^\lambda U={\textstyle\bigwedge}^tU\) if \(\lambda=(1,\ldots,1)\vdash t\). A Schur module \(S^\lambda U\) is nonzero if and only if \(\lambda_{\dim U}\neq 0\), and in this case it is irreducible over \(\operatorname{GL}(U)\), which means that it does not carry a proper nontrivial subrepresentation.

A partition \(\lambda \vdash t\) is typically identified with a Young diagram consisting of \(t\) boxes. These boxes are arranged in a matrix format with one box placed at the \((i,j)\)-entry for each \(i\geq1\) and \(1\leq j\leq\lambda_i\). For example, if \(\lambda=(2,1,1)\), then \[\label{Eqn:Y211} \ydiagram{2,1,1}\tag{1}\] is the corresponding Young diagram. Transposing the Young diagram of a given partition \(\lambda\), we obtain the conjugate partition \(\lambda'\) of \(\lambda\). For the example above we have \(\lambda'=(3,1)\) with the following Young diagram. \[\label{Eqn:Y31} \ydiagram{3,1}\tag{2}\] For our purpose hook diagrams are of special interest. They are Young diagrams that have at most one box in all but the first row, that is, \(\lambda_2\leq1\), such as 1 and 2 . Hook diagrams are related to Koszul differentials via \[Z_{a,b}(U)=S^{(b,1,\ldots,1)}U\quad\text{and}\quad Z^{a,b}(U)=S^{(a+1,1,\ldots,1)}U\] for the partitions \((b,1,\ldots,1)\) and \((a+1,1,\ldots,1)\) of \(a+b\).

A fundamental topic in representation theory is to describe the induced \(\operatorname{GL}(U)\times\operatorname{GL}(W)\)-module structure of a given representation of \(\operatorname{GL}(U\otimes W)\). Basic formulas along this line are as follows.

Consider the identification \[\begin{array}{rcl} T^{q+1}U\otimes T^{q+1}W & \xlongrightarrow{\cong} & T^{q+1}(U\otimes W), \\ (u_0 \otimes\cdots\otimes u_q)\otimes(w_0\otimes\cdots\otimes w_q) & \longmapsto & (q+1)!(u_0\otimes w_0)\otimes\cdots\otimes(u_q\otimes w_q). \end{array}\] Then its restriction to \({\textstyle\bigwedge}^{q+1}U\otimes{\textstyle\bigwedge}^{q+1}W\), under the averaging embeddings \({\textstyle\bigwedge}^{q+1}U\subset T^{q+1}U\) and \({\textstyle\bigwedge}^{q+1}W\subset T^{q+1}W\), is nothing but the determinant map \[\det:{\textstyle\bigwedge}^{q+1}U\otimes{\textstyle\bigwedge}^{q+1}W\to S^{q+1}(U\otimes W)\] for which if \((s_0\wedge\cdots\wedge s_q)\otimes(t_0\wedge\cdots\wedge t_q)\in{\textstyle\bigwedge}^{q+1}U\otimes{\textstyle\bigwedge}^{q+1}W\) is viewed as a matrix \((x_{i,j})\) with entries \(x_{i,j}:=s_i\otimes t_j\in U\otimes W\), then \[\det(x_{i,j})=\sum_{\sigma\in\mathfrak{S}_{q+1}}(\operatorname{sgn}\sigma)x_{\sigma(0),0}\cdots x_{\sigma(q),q}.\] Compared to it another restriction, denoted by \[\operatorname{edet}:S^{q+1}U\otimes{\textstyle\bigwedge}^{q+1}W\to{\textstyle\bigwedge}^{q+1}(U\otimes W),\] receives much less attention, but it actually appears in the literature under the name “exterior minor" [21] or”permanent." For a given form \((s_0\cdots s_q)\otimes(t_0\wedge\cdots\wedge t_q)\in S^{q+1}U\otimes{\textstyle\bigwedge}^{q+1}W\), the map \(\operatorname{edet}\) sends the matrix \(x_{i,j}:=s_i\otimes t_j\) to \[\operatorname{edet}(x_{i,j})=\sum_{\sigma\in\mathfrak{S}_{q+1}}x_{\sigma(0),0}\wedge\cdots\wedge x_{\sigma(q),q}.\]

The maps \(\det\) and \(\operatorname{edet}\) enjoy the so-called Jacobi’s formula. Note that for fixed bases \(u_0,u_1,\ldots\) and \(w_0,w_1,\ldots\) of \(U\) and \(W\), respectively, the tensors \(u_i\otimes w_j\) form a basis of \(U\otimes W\) so that its dual basis \(\partial/\partial(u_i\otimes w_j)\) acts as partial differentials on both \(S^\ast(U\otimes W)\) and \({\textstyle\bigwedge}^\ast(U\otimes W)\).

The following property of \(\det\) and \(\operatorname{edet}\) is used later in the proof of 36.

We introduce an operation that serves as a fundamental tool for establishing interesting equations, as in 17 below, along with its analogue over exterior algebras.

The following lemma is needed for the proof of a main theorem.

2.2 Higher secant varieties to higher osculating varieties↩︎

Let \(z\in\operatorname{Sm}X\) be a smooth point of \(X\subseteq\mathbb{P}V\) and \(n=\dim X\). Take a smooth lifting \[\label{Eqn:LocalPara} \phi=\phi(t)=\phi(t_1,\ldots,t_n):(\mathbb{C}^n,0)\to (\widehat{X},\widehat{z})\subseteq(V^\ast,\widehat{z})\tag{3}\] of a local parametrization of \(X\) at \(z\), that is, passing through the projection map \((V^\ast,\widehat{z})\to(\mathbb{P}V,z)\), we obtain a local parametrization of the germ \((X,z)\). Then for an integer \(k\geq 0\) the projective \(k\)-osculating space to \(X\subseteq\mathbb{P}V\) at \(z\) is defined as \[\mathbb{T}^k_zX=\mathbb{P}\left\langle\frac{\partial^{\alpha_1+\cdots+\alpha_n}}{\partial t_1^{\alpha_1}\cdots\partial t_n^{\alpha_n}}\phi(0)\in V^\ast:\alpha_1+\cdots+\alpha_n\leq k\right\rangle^\ast\subseteq\mathbb{P}V.\] For example, if \(k=0\), then \(\mathbb{T}^0_zX\) is the point \(\{z\}\), if \(k=1\), then \(\mathbb{T}^1_zX\) is the projective tangent space \(\mathbb{T}_zX\) to \(X\) at \(z\), and if \(k=2\), then \(\mathbb{T}^2_zX\) is the projective osculating space to \(X\) at \(z\).

Then our objects of interest are higher secant varieties to higher osculating varieties. For integers \(q\geq 1\) and \(k\geq0\) the \(q\)-secant variety to the \(k\)-osculating variety to \(X\subseteq\mathbb{P}V\) is defined to be the Zariski closure \[\sigma_q\tau^kX=\overline{\bigcup_{z_1,\ldots,z_q\in\operatorname{Sm}X}\langle\mathbb{T}^k_{z_1}X,\ldots,\mathbb{T}^k_{z_q}X\rangle}\subseteq\mathbb{P}V,\] where \((z_1,\ldots,z_q)\in(\operatorname{Sm}X)^q\) runs over all general \(q\)-tuples of smooth points in \(X\) so that \(\dim\langle\mathbb{T}^k_{z_1}X,\ldots,\mathbb{T}^k_{z_q}X\rangle\) attains the maximum, and \(\langle-\rangle\) means the linear span in \(\mathbb{P}V\). If \(q=1\), then \(\tau^kX:=\sigma_1\tau^kX\) is the \(k\)-osculating variety to \(X\subseteq\mathbb{P}V\), and if in addition \(k=1\), then \(\tau X:=\tau^1X\) is the tangent variety to \(X\subseteq\mathbb{P}V\). If \(k=0\), then \(\sigma_qX:=\sigma_q\tau^0X\) is the \(q\)-secant variety to \(X\subseteq\mathbb{P}V\). Note that \(\sigma_q\tau^kX=\sigma_q(\tau^kX)\).

The problem of finding equations of \(\sigma_qX\subseteq\mathbb{P}V\) has been extensively studied, carrying hugely numerous results. Among them we may consider the following as a starting point.

1. \(I(\sigma_qX)\) does not have any nonzero \(q\)-form.

2. Its \((q+1)\)-forms can be computed via the prolongation.

As for equations of tangent varieties, Oeding and Raicu have focused on the case of Segre-Veronese varieties \[Y:=\mathbb{P}V_1\times\cdots\times\mathbb{P}V_\ell\subseteq\mathbb{P}(S^{d_1}V_1\otimes\cdots\otimes S^{d_\ell}V_\ell).\]

Furthermore, [18] explicitly describes the minimal generators of \(I(\tau Y)\) for any Segre-Veronese variety \(Y\). Accordingly, \(I(\tau Y)_2\) has the form \[I(\tau Y)_2=\bigoplus_{\lambda^1,\ldots,\lambda^\ell}S^{\lambda^1}V_1\otimes\cdots\otimes S^{\lambda^\ell}V_\ell,\] where \(\lambda^1\vdash 2d_1,\ldots,\lambda^\ell\vdash 2d_\ell\) run over all partitions of the forms \[\lambda^i=(2d_i-a_i\geq a_i)\quad\text{with}\quad a_1+\cdots+a_\ell\in\{4,6,8,\ldots\}.\] For example, if \(Y=\mathbb{P}V_1\times\cdots\times\mathbb{P}V_4\subseteq\mathbb{P}(V_1\otimes\cdots\otimes V_4)\), then \[\label{Eqn:4FactorQuadric} I(\tau Y)_2={\textstyle\bigwedge}^2V_1\otimes\cdots\otimes{\textstyle\bigwedge}^2V_4.\tag{4}\] The work of Oeding and Raicu settles a conjecture of Landsberg and Weyman on generators of \(I(\tau Y)\) for Segre varieties \(Y\), that is, \(d_1=\cdots=d_\ell=1\).

We end this subsection with an observation about the equations of the smallest possible degree for \(\sigma_q\tau^kX\subseteq\mathbb{P}V\). Let \(\emptyset\neq U\subseteq\operatorname{Sm}X\) be any open dense subset that maximizes \(\dim\mathbb{T}^k_zX\) for all \(z\in U\), and put \(L=\mathcal{O}_X(1)|_U\). Then we may view \(S^{q+1}V\) as a subspace of \(T^{q+1}H^0(L)=H^0(U^{q+1},L^{\boxtimes q+1})\), where \(L^{\boxtimes q+1}\) is the box product \(L\boxtimes\cdots\boxtimes L\) on the \((q+1)\)-th Cartesian power \(U^{q+1}\). Consider the big diagonal \(\Delta^{q+1}:=\bigcup_{0\leq i<j\leq q}\Delta_{i,j}\subset U^{q+1}\), where \(\Delta_{i,j}:=\{(z_0,\ldots,z_q)\in U^{q+1}:z_i=z_j\}\). Recall that the \(\ell\)-th symbolic power \(\mathcal{I}_{\Delta^{q+1}/U^{q+1}}^{(\ell)}\) is the ideal sheaf of all functions having vanishing order \(\geq\ell\) along \(\Delta^{q+1}\) and that \(\mathcal{I}_{\Delta^2/U^2}^{(\ell)}\) coincides with the \(\ell\)-th ordinary power \(\mathcal{I}_{\Delta^2/U^2}^l\) since \(U\) is smooth.

19 reveals a geometric meaning of 4 and applies extensively to higher secant varieties to the \(k\)-osculating varieties to \((2k+2)\)-factor Segre varieties.

2.3 Syzygies↩︎

Let \(S=\mathbb{C}[x_0,\ldots,x_r]\) be a polynomial ring in \(r+1\) variables with the standard grading and \(M\) be a finitely generated graded \(S\)-module. Consider its minimal graded free resolution \[\begin{tikzcd} F_0 & \ar[l,"d_1",swap] F_1 & \ar[l,"d_2",swap] \cdots & \ar[l,"d_i",swap] F_i & \ar[l,"d_{i+1}",swap] \cdots \end{tikzcd}\] so that \(\operatorname{coker}\,d_1\cong M\), and \(\operatorname{im}d_i\subseteq(x_0,\ldots,x_r)\cdot F_{i-1}\) for all \(i\geq 1\), and write \[F_i=\bigoplus_{j\in\mathbb{Z}}K_{i,j}(M)\otimes S(-i-j)\] by grouping the minimal homogeneous generators of each degree. Then the vector space \(K_{i,j}(M)\) is regarded as the space of \(i\)-th syzygies of weight \(j\) of \(M\). It is isomorphic to the cohomology group of the Koszul type complex \[\begin{tikzcd} {\textstyle\bigwedge}^{i+1}V\otimes M_{j-1} \ar[r] & {\textstyle\bigwedge}^{i}V\otimes M_j \ar[r] & {\textstyle\bigwedge}^{i-1}V\otimes M_{j+1} \end{tikzcd}\] at the middle, where \(V=S_1\) is the space of linear forms in \(S\), and we call it the \((i,j)\)-th Koszul cohomology group of \(M\).

For our study, of special interest are syzygies of the minimal possible weight of \(I(\sigma_q\tau^kX)\). Recall that \(I(\sigma_q\tau^kX)_q=0\) by the nature of higher secant varieties.

As a fundamental example syzygies of higher secant varieties to Segre varieties have been well understood.

Consequently, the bottom syzygies of \(\sigma_qY\) for a \(2\)-factor Segre variety \(Y\) form the space \[K_{p,q+1}(I(\sigma_qY))=\bigoplus_{a+b=p}S^{(a+1,1,\ldots,1)}V_1\otimes S^{(b+1,1,\ldots,1)}V_2\] for partitions \((a+1,1,\ldots,1)\) and \((b+1,1,\ldots,1)\) of \(p+q+1\).

We often use Betti tables to encapsulate minimal free resolutions and Koszul cohomology groups. When we restrict ourselves to the case of \(I(\tau Y)\) for \(4\)-factor Segre varieties \(Y\), they are diagrams whose entries are \(\beta_{i,j}:=\dim K_{i,j}(I(\tau Y))\). In the Betti table the column \(i=0\) is controlled by Oeding-Raicu’s result, whereas the row \(j=2\) is determined by one of our main results: \[\begin{array}{c|ccccl} j\setminus i & 0 & 1 & 2 & 3 & \cdots \\ \hline 2 & \beta_{0,2} & \beta_{1,2} & \beta_{2,2} & \beta_{3,2} & \leftarrow\mathrm{\Cref{Thm:SegreDecompo}} \\ 3 & \beta_{0,3} & \beta_{1,3} & \beta_{2,3} & \beta_{3,3} & \multirow{2}{*}{\cdots} \\ 4 & \beta_{0,4} & \beta_{1,4} & \beta_{2,4} & \beta_{3,4} & \\ 5 & \beta_{0,5} & \beta_{1,5} & \beta_{2,5} & \beta_{3,5} & \\ \vdots & \uparrow & \multicolumn{2}{c}{\vdots} \\ & \cite{MR3240996} & \phantom{\cite{MR3240996}} & \phantom{\cite{MR3240996}} & \phantom{\cite{MR3240996}} \end{array}\]

2.4 Tensors of linear forms↩︎

This subsection presents the concept of a tensor of linear forms. Recall that an element of a given tensor product \(V_1\otimes\cdots\otimes V_\ell\), namely an \(\ell\)-way tensor, is called simple if it is equal to \(v_1\otimes\cdots\otimes v_\ell\) for some nonzero vectors \(0\neq v_i\in V_i\), \(1\leq i\leq\ell\).

Here, \(1\)-generic matrices of linear forms expand the discussion of generic matrices, that is, the matrices whose entries are pairwise distinct indeterminates.

The \(X\)-simplicity can be understood in an alternative way.

2.5 Symmetric powers of smooth curves↩︎

Throughout this subsection we assume \(C\) to be a smooth projective curve of genus \(g\) and discuss line bundles and divisors on the \((q+1)\)-th symmetric power \(C_{q+1}\) of \(C\), that is, the quotient \[C_{q+1}=C^{q+1}/\mathfrak S_{q+1}\] of the \((q+1)\)-th Cartesian power \(C^{q+1}\) of \(C\) by the factor-permuting action of \(\mathfrak S_{q+1}\) on \(C^{q+1}\). It is in fact the Hilbert scheme of effective divisors \(\xi\) of degree \(q+1\) on \(C\), being a smooth projective variety of dimension \(q+1\).

We list notions and facts about \(C_{q+1}\). Let \(B\in\operatorname{Pic}(C)\) stand for a line bundle on \(C\).

1. Write \(B^{\boxtimes q+1}=B\boxtimes\cdots\boxtimes B\) for the \((q+1)\)-th box power of \(B\) on \(C^{q+1}\).

2. We denote by \(S_{q+1,B}\) the line bundle on \(C_{q+1}\) to which \(B^{\boxtimes q+1}\) descends with respect to the canonical \(\mathfrak S_{q+1}\)-linearization. For a point \(z\in C\) the effective divisor \(X_{q+1}^z:=\{\xi\in C_{q+1}:\xi\ni z\}\) lies in \(|S_{q+1,\mathcal{O}_C(z)}|\).

3. Let \(a_{q+1}:C\times C_q\to C_{q+1}\) be the addition map \((z,\xi)\mapsto z+\xi\). It acts as the universal family of degree \(q+1\) effective divisors on \(C\).

4. The pushforward \(E_{q+1,B}:=a_{q+1,\ast}(B\boxtimes\mathcal{O}_{C_q})\) is a vector bundle of rank \(q+1\) on \(C_{q+1}\) whose fiber over each \(\xi\in C_{q+1}\) is \(H^0(B|_\xi)\).

5. Put \(N_{q+1,B}=\det E_{q+1,B}\). Its global sections are \(H^0(N_{q+1,B})={\textstyle\bigwedge}^{q+1}H^0(B)\), and furthermore, \(H^i(N_{q+1,B})={\textstyle\bigwedge}^{q+1-i}H^0(B)\otimes S^iH^1(B)\) for all \(0\leq i\leq q+1\).

6. Nonreduced effective divisors of degree \(q+1\) on \(C\) form an effective divisor on \(C_{q+1}\), namely the diagonal divisor \(\Delta_{q+1}\subset C_{q+1}\).

7. There is a divisor \(\delta_{q+1}\) on \(C_{q+1}\) such that \(2\delta_{q+1}\) is linearly equivalent to \(\Delta_{q+1}\). Then \(N_{q+1,B}\cong S_{q+1,B}(-\delta_{q+1})\).

8. We have the natural isomorphism \(\Delta_2\cong C\) under which \(\mathcal{O}_{C_2}(-\delta_2)|_{\Delta_2}\cong\omega_C\).

9. The product \(C_{q+1}\times C^i\) has an effective divisor \[D_{q+1}^i:=\{(\xi,z_1,\ldots,z_i):\xi\ni z_j\mathrm{ for some }1\leq j\leq i\}.\] Its fibers are given as \[\begin{align} (D_{q+1}^i)_\xi & =\xi\times C^{i-1}+\cdots+C^{i-1}\times\xi\in|\mathcal{O}_C(\xi)^{\boxtimes i}|\quad\mathrm{and} \\ (D_{q+1}^i)_{(z_1,\ldots,z_i)} & =X_{q+1}^{z_1}+\cdots+X_{q+1}^{z_i}\in|S_{q+1,\mathcal{O}_C(z_1+\cdots+z_i)}|. \end{align}\] Also, \(C^i\) has a similar effective divisor \[D^{i-1,1}:=\{(z_1,\ldots,z_{i-1},z_i)\in C^i:z_j=z_i\mathrm{ for some }1\leq j\leq i-1\}.\]

For these matters see [27], [28], [29], [30], and [5].

Let \(B\in\operatorname{Pic}(C)\) be a line bundle on \(C\) and \(q\geq0\) be an integer. Suppose that \(B\) is \(q\)-very ample so that \(E_{q+1,B}\) is globally generated. Let \(M_{q+1,B}\) be the kernel bundle in the short exact sequence \[\begin{tikzcd} 0 \ar[r] & M_{q+1,B} \ar[r] & H^0(B)\otimes\mathcal{O}_{C_{q+1}} \ar[r] & E_{q+1,B} \ar[r] & 0. \end{tikzcd}\] Taking \({\textstyle\bigwedge}^{q+1}\) we have a locally free resolution \[\label{Eqn:NResolution} \begin{tikzcd} F_{q+1}(B)_i:=S^iM_{q+1,B}\otimes{\textstyle\bigwedge}^{q+1-i}H^0(B) \ar[r] & \cdots \ar[r] & F_{q+1}(B)_1 \ar[r] & F_{q+1}(B)_0 \end{tikzcd}\tag{6}\] of \(\det E_{q+1,B}=N_{q+1,B}\).

For the proof of 4 we need to compute some sheaf cohomology groups related to the resolutions \(F_{q+1}(B)_\ast\) and an \((i-1)\)-very ample line bundle \(L_1\) on \(C\) for an integer \(i\geq 1\), which employs the pullback \[Q^i_{L_1}:=\pi^\ast_iM_{i,L_1}\] via the quotient map \(\pi_i:C^i\to C_i\). Notice that \(Q^1_{L_1}\) is the well-known kernel bundle \(M_{L_1}:=\ker(H^0(L_1)\otimes\mathcal{O}_C\to L_1)\) and that the short exact sequence \[\label{Eqn:RathmannSES} \begin{tikzcd} 0 \ar[r] & Q^i_{L_1} \ar[r] & Q^{i-1}_{L_1}\boxtimes\mathcal{O}_C \ar[r] & \mathcal{O}_{C^{i-1}}\boxtimes L_1(-D^{i-1,1}) \ar[r] & 0 \end{tikzcd}\tag{7}\] [31] holds. For convenience in the following a line bundle \(B\) on \(C\) is called Clifford if \[B\in \begin{cases} \emptyset & \mathrm{if g=0} \\ \{\mathcal{O}_C,\omega_C\} & \mathrm{if C is not hyperelliptic with g\geq 1} \\ \{G^m:0\leq m\leq g-1\} & \mathrm{if C is hyperelliptic with |G|=g^1_2}. \end{cases}\]

We close this subsection with a computation of higher direct images, which enables us to argue, via the Leray spectral sequence, a kind of sheaf cohomology vanishing around the incidence divisor \(D_{q+1}^i\subset C_{q+1}\times C^i\).

3 Green-Lazarsfeld classes revisited↩︎

In this section we reformulate the Green-Lazarsfeld classes using multilinear algebra. This approach sheds light on their useful substantial structures.

We begin with a preparatory step. In what follows for simplicity we use the Einstein summation convention: “repeated indices are implicitly summed over." For example, we may write \[\label{Eqn:fG} f=\frac{1}{\alpha!}u^\alpha\otimes f_\alpha\quad\text{and}\quad G=\frac{1}{\beta!}w^\beta\otimes G_\beta,\tag{8}\] where \(|\alpha|=a\) and \(|\beta|=b+1\), for arbitrary elements \(f\in S^aU\otimes{\textstyle\bigwedge}^{b+q+1}U\) and \(G\in S^{b+1}W\otimes{\textstyle\bigwedge}^{a+q}W\) with respect to given bases \(u\) and \(w\) of \(U\) and \(W\), respectively. Also, we ignore the scaling of nonzero coefficients when it causes no issues.

The products \(\boxtimes^0\) and \(\boxtimes_0\) have the following key properties in relation to some subspaces of \(K_{a+b,q+1}(U,W)\) and \(K^{a+b,q+1}(U,W)\).

These features give rise to new products derived from the previous ones.

Note that \(K_{p,q+1}(U,W)\) is nothing but \(K_{p,q+1}(I(\sigma_qY))\) for the Segre variety \(Y:=\mathbb{P}U\times\mathbb{P}W\subseteq\mathbb{P}(U\otimes W)\).

We now conclude this section with an interpretation of the Green–Lazarsfeld classes, which also provides a concrete description of the part \(m=1\) of the Lascoux resolutions (25).

4 Proofs↩︎

In this section we prove the main results.

4.1 Proof of 7↩︎

More generally, we show that \[K_{p,q+1}(V_1,\ldots,V_\ell)=\bigoplus_{p_\ast\vdash p}B_{p_\ast,q+1}(V_1,\ldots,V_\ell)\] for every \(\ell\geq 1\), where \[K_{p,q+1}(V_1,\ldots,V_\ell):=\begin{cases} Z_{p,q+1}(V_1\otimes\cdots\otimes V_\ell)\cap\left({\textstyle\bigwedge}^p(V_1\otimes\cdots\otimes V_\ell)\otimes({\textstyle\bigwedge}^{q+1}V_1\otimes\cdots\otimes{\textstyle\bigwedge}^{q+1}V_\ell)\right) & \mathrm{if }2\mid\ell \\ Z^{p,q+1}(V_1\otimes\cdots\otimes V_\ell)\cap\left(S^p(V_1\otimes\cdots\otimes V_\ell)\otimes({\textstyle\bigwedge}^{q+1}V_1\otimes\cdots\otimes{\textstyle\bigwedge}^{q+1}V_\ell)\right) & \mathrm{if }2\nmid\ell. \end{cases}\] This suffices because of 22 which also shows [ThmItem:SegreDecompoTrivial].

We proceed by induction on \(\ell\geq 1\). For the initial step \(\ell=1\) there is nothing to prove. Set \(\ell\geq 2\), and for simplicity assume that \(\ell\) is even; the odd case is parallel to the even one. Since \({\textstyle\bigwedge}^{q+1}V_1\otimes\cdots\otimes{\textstyle\bigwedge}^{q+1}V_{\ell-1}\otimes{\textstyle\bigwedge}^{q+1}V_\ell\subseteq{\textstyle\bigwedge}^{q+1}(V_1\otimes\cdots\otimes V_{\ell-1})\otimes{\textstyle\bigwedge}^{q+1}V_\ell\), we have \[K_{p,q+1}(V_1,\ldots,V_{\ell-1},V_\ell)\subseteq K_{p,q+1}(V_1\otimes\cdots\otimes V_{\ell-1},V_\ell)= \bigoplus_{a+b=p}Z^{a,b+q+1}(V_1\otimes\cdots\otimes V_{\ell-1})\otimes Z^{b,a+q+1}(V_\ell)\] by 39 up to the products \(\boxtimes\). Let \(M\subseteq K_{p,q+1}(V_1,\ldots,V_{\ell-1},V_\ell)\) be an irreducible submodule over \(\operatorname{GL}(V_1)\times\cdots\times\operatorname{GL}(V_{\ell-1})\times\operatorname{GL}(V_\ell)\). Obviously, it has the form \[M=M'\otimes Z^{b,a+q+1}(V_\ell)\] with \(M'\subseteq Z^{a,b+q+1}(V_1\otimes\cdots\otimes V_{\ell-1})\) irreducible over \(\operatorname{GL}(V_1)\times\cdots\times\operatorname{GL}(V_{\ell-1})\). Remind that \(M'\otimes Z^{b,a+q+1}(V_\ell)\subseteq{\textstyle\bigwedge}^p(V_1\otimes\cdots\otimes V_{\ell-1}\otimes V_\ell)\otimes({\textstyle\bigwedge}^{q+1}V_1\otimes\cdots\otimes{\textstyle\bigwedge}^{q+1}V_{\ell-1}\otimes{\textstyle\bigwedge}^{q+1}V_\ell)\). 39[Thm:GreenLazarsfeldItem1] and 16 imply that \[\begin{align} M' & \subseteq Z^{a,b+q+1}(V_1\otimes\cdots\otimes V_{\ell-1})\cap\left(S^a(V_1\otimes\cdots\otimes V_{\ell-1})\otimes({\textstyle\bigwedge}^{b+q+1}V_1\otimes\cdots\otimes{\textstyle\bigwedge}^{b+q+1}V_{\ell-1})\right) \\ & = K_{a,b+q+1}(V_1,\ldots,V_{\ell-1}). \end{align}\] That is, we reach \[K_{p,q+1}(V_1,\ldots,V_{\ell-1},V_\ell)\subseteq\bigoplus_{a+b=p}K_{a,b+q+1}(V_1,\ldots,V_{\ell-1})\otimes Z^{b,a+q+1}(V_\ell).\] On the other hand, by 39[Thm:GreenLazarsfeldItem2] each summand indeed appears in \(K_{p,q+1}(V_1,\ldots,V_{\ell-1},V_\ell)\). In conclusion, we have a decomposition \[K_{p,q+1}(V_1,\ldots,V_{\ell-1},V_\ell)=\bigoplus_{a+b=p}K_{a,b+q+1}(V_1,\ldots,V_{\ell-1})\otimes Z^{b,a+q+1}(V_\ell),\] making the induction work.

To verify 7[ThmItem:SegreDecompoUpper] we assume that \(B_{p_\ast,q+1}(V_1,\ldots,V_{2k+2})\neq 0\) for some ordered partition \(p_\ast=(p_1,\ldots,p_{2k+2})\vdash p\), that is, the inequalities in Condition ?? all hold. Sum them up to obtain \[\sum_{i=1}^{2k+2}\dim V_i\geq (2k+1)p+(2k+2)(q+1).\] In other words, if \((2k+1)p>\sum_i(\dim V_i-q-1)\), then \(B_{p_\ast,q+1}(V_1,\ldots,V_{2k+2})=0\) always.

4.2 Proofs of [Thm:Main1,,Thm:Main2]↩︎

We provide a single proof that covers both [Thm:Main1,,Thm:Main2]. We begin with the following commutative diagram. \[\begin{tikzcd} X \ar[r,dashed,"T^\ast|_X"] \ar[d,phantom,"{\rotatebox[origin=c]{-90}{\subseteq}}"] & \mathbb{P}V_1\times\cdots\times\mathbb{P}V_{2k+2} \ar[d,phantom,"{\rotatebox[origin=c]{-90}{\subseteq}}"] \\ \mathbb{P}V \ar[r,dashed,"T^\ast",swap] & \mathbb{P}(V_1\otimes\cdots\otimes V_{2k+2}) \end{tikzcd}\] Note that the bottom map \(T^\ast:\mathbb{P}V\dashrightarrow\mathbb{P}(V_1\otimes\cdots\otimes V_{2k+2})\) is linear. Thus, pulling back bottom syzygies of \(\sigma_q\tau^kY\) for \(Y:=\mathbb{P}V_1\times\cdots\times\mathbb{P}V_{2k+2}\subseteq\mathbb{P}(V_1\otimes\cdots\otimes V_{2k+2})\), we obtain those of \(\sigma_q\tau^kX\subseteq\mathbb{P}V\): \[b_{p_\ast,q+1}(T):B_{p_\ast,q+1}(V_1,\ldots,V_{2k+2})\to K_{p,q+1}(I(\sigma_q\tau^kX)),\] and such bottom syzygies can be nontrivial as ensured by 39[Thm:GreenLazarsfeldItem2]. To be more specific, if for each \(1\leq i\leq 2k+2\) we take an appropriate basis \(u\) of \(V_i\) and put \[f_i=u_0^{p_i}\otimes u_{\{0,\ldots,p-p_i+q\}}\in Z^{p_i,p-p_i+q+1}(V_i),\] then \(f_1\otimes\cdots\otimes f_{2k+2}\) maps to a nonzero bottom syzygy in \(K_{p,q+1}(I(\sigma_q\tau^kX))\) through the products \(\boxtimes\).

4.3 Proof of 4↩︎

In the spirit of 19 we find that \[I(\sigma_q\tau^kC)_{q+1}=H^0(S_{q+1,L}(-(2k+2)\delta_{q+1})).\] (One could examine the short exact sequences \[\begin{tikzcd} 0 \ar[r] & S_{2,L}(-(i+1)\Delta_2) \ar[r] & S_{2,L}(-i\Delta_2) \ar[r] & L^2\otimes\omega_{\Delta_2}^{2i} \ar[r] & 0 \end{tikzcd}\] for the sake of certainty.) So we show that for every integer \(\ell\geq 2\) if \(\deg L\geq\ell(2g+q)\), then a multiplication map \[H^0(N_{q+1,L_1})\otimes\cdots\otimes H^0(N_{q+1,L_\ell})\to H^0(S_{q+1,L}(-\ell\delta_{q+1}))\] is surjective for some factorization \(L_1\otimes\cdots\otimes L_\ell=L\) in \(\operatorname{Pic}(C)\).

Suppose that \(\deg L\geq\ell(2k+q)\). We may decompose \(L\) as \[L=L_1\otimes\cdots\otimes L_\ell\] in \(\operatorname{Pic}(C)\) so that

1. \(\deg L_i\geq 2g+q\) for all \(1\leq i\leq\ell\), and

2. if \(g\geq 1\), and if \(\deg L_1=\deg L_2=2g+q\), then \(L_1\) is not isomorphic to \(L_2\).

Indeed, let \(L_3,\ldots,L_\ell\) be any line bundles of degree \(\geq 2g+q\) such that \(L_{1,2}:=L\otimes L_3^{-1}\otimes\cdots\otimes L_\ell^{-1}\) has degree \(\geq 4g+2q\). Now if \(\deg L_{1,2}>4g+2q\), then choose an arbitrary line bundle \(L_2\) of degree \(2g+q\leq\deg L_2\leq\deg L_{1,2}-(2g+q)\), but if \(\deg L_{1,2}=4g+2q\), then pick a general line bundle \(L_2\) of degree \(2g+q\) so that \(L_2^2\not\cong L_{1,2}\) in the case \(g\geq1\). Our procedure ends with \(L_1:=L_{1,2}\otimes L_2^{-1}\).

For such a choice of \(L_1,\ldots,L_\ell\) let \(F_{q+1}(\widehat{L_1})_\ast\) be the total complex of \(F_{q+1}(L_2)_\ast\otimes\cdots\otimes F_{q+1}(L_\ell)_\ast\) (see 6 ), that is, \[F_{q+1}(\widehat{L_1})_i=\bigoplus_{i_2+\cdots+i_\ell=i}S^{i_2}M_{q+1,L_2}\otimes\cdots\otimes S^{i_\ell}M_{q+1,L_\ell}\] with constant bundle factors omitted, and \(F_{q+1}(\widehat{L_1})_0=H^0(N_{q+1,L_2})\otimes\cdots\otimes H^0(N_{q+1,L_\ell})\). Observe that \(N_{q+1,L_1}\otimes F_{q+1}(\widehat{L_1})_\ast\) is a resolution of \(N_{q+1,L_1}\otimes N_{q+1,L_2}\otimes\cdots\otimes N_{q+1,L_\ell}=S_{q+1,L}(-\ell\delta_{q+1})\) and that \(H^0(N_{q+1,L_1}\otimes F_{q+1}(\widehat{L_1})_0)=H^0(N_{q+1,L_1})\otimes H^0(N_{q+1,L_2})\otimes\cdots\otimes H^0(N_{q+1,L_\ell})\). Therefore, it is enough to obtain the vanishing \[H^i(N_{q+1,L_1}\otimes F_{q+1}(\widehat{L_1})_i)=0\] for every \(i\geq 1\).

Since \(S^{i_j}M_{q+1,L_j}\) is a direct summand of \(T^{i_j}M_{q+1,L_j}\) for each \(2\leq j\leq \ell\), the desired vanishing in turn follows from the vanishing \[\label{Eqn:VanishingLB} H^i(N_{q+1,L_1}\otimes M_{q+1,B_1}\otimes\cdots\otimes M_{q+1,B_i})=0\tag{10}\] for line bundles \(B_1,\ldots,B_i\) of degree \(\geq 2g+q\) such that if \(g\geq1\), and if \(\deg L_1=\deg B_1=2g+q\), then \(L_1\not\cong B_1\). Using the Leray spectral sequences for the projection maps \[\operatorname{pr}_1:C_{q+1}\times C^i\to C_{q+1}\quad\mathrm{and}\quad\operatorname{pr}_2:C_{q+1}\times C^i\to C^i\] in conjunction with 32, we find that 10 is equivalent to the vanishing \[H^i({\textstyle\bigwedge}^{q+1}Q^i_{L_1}\otimes B_1\boxtimes\cdots\boxtimes B_i)=0.\] Applying 31 we are done.

4.4 Proof of 5↩︎

By 1 the tangent variety \(\tau X\) has a quadratic equation, but \(\sigma_2X\) can not by its nature. Thus, \(\tau X\) is strictly included in \(\sigma_2X\), which means that \[\dim\tau X=2\dim X\quad\text{and}\quad\dim\sigma_2X=2\dim X+1\] due to [6].

5 Comments and questions↩︎

This section collect comments and questions on this study.

5.1 Tangent varieties to smooth curves↩︎

Let \(C\) be a smooth projective curve of genus \(g\) and \(C\subseteq\mathbb{P}H^0(L)\) be a complete embedding by a very ample line bundle \(L\) on \(C\).

This subsection focuses on the quantity \[\ell(\tau C):=\max\{p\geq 0:K_{p,2}(I(\tau C))\neq 0\}.\]

The rational case \(g=0\) has been thoroughly studied and understood, which indeed forms the core of the result by Aprodu–Farkas–Papadima–Raicu–Weyman mentioned in the introduction.

As a corollary, a sophisticated computation of what are known as the Koszul modules yields \[\ell(\tau C)=\lfloor d/2\rfloor-2\] when \(C\) is rational.

With regard to nonrational curves \(C\), 6 gives rise to the lower bound \[\label{Eqn:1473LB} \ell(\tau C)\geq\left\lfloor\frac{d-\operatorname{gon}(C)}{3}\right\rfloor-g-1\tag{11}\] when \(L\) has degree \(d\geq 3g+\operatorname{gon}(C)+3\), where \(\operatorname{gon}(C)\) is the gonality of \(C\). Indeed, let \(p\) be the right-hand side value in 11 , and choose line bundles \(L_1,\ldots,L_4\) on \(C\) so that

1. \(L_1\) is a line bundle computing the gonality, and

2. \(L_2,L_3,L_4\) have degree \(\geq g+p+1\) under the constraint that \(L_2\otimes L_3\otimes L_4=L\otimes L_1^{-1}\).

Then since \[h^0(L_i)\geq \begin{cases} 2 & \mathrm{if }i=1 \\ p+2 & \mathrm{if }2\leq i\leq 4, \end{cases}\] Condition ?? holds for the ordered partition \(p_\ast=(p,0,0,0)\vdash p\). In addition, it is conjectured in [32] that \[\ell(\tau C)\leq\left\lfloor\frac{d-3g}{2}\right\rfloor-2,\] which is based on some computational evidence.

This discussion brings us to the following question.

One can find that if \(\deg L\geq (2k+1)(g+p+q)+\operatorname{gon}^q(C)\) for an integer \(p\geq0\), where \(\operatorname{gon}^q(C):=\min\{d\geq 1:C\mathrm{ carries a }g_d^q\}\), then \(K_{p,q+1}(I(\sigma_q\tau^kC))\neq0\).

On the other hand, let us look at the initial stage of the minimal free resolution of \(I(\tau C)\). Due to 4 we have already found the source of all the degree \(q+1\) equations of \(\sigma_q\tau^kC\), but in order to complete the full matryoshka picture of Eisenbud–Koh–Stillman’s work, we need to understand not merely the single component \(I(\sigma_q\tau^kC)_{q+1}\), but all components of the ideal.

Park made a guess [32] that it is the case for \(\tau C\subseteq\mathbb{P}H^0(L)\) if \[\deg L\geq 5g+5,\] and more generally, if \(\deg L\geq 4g+3+(g+2)p\) for an integer \(p\geq 1\), then \(\tau C\subseteq\mathbb{P}H^0(L)\) satisfies property \(N_p\), which means that \(K_{i,j}(I(\tau C))=0\) whenever \(i\leq p-1\) and \(j\geq 3\).

5.2 Syzygies over exterior algebras↩︎

The theory of syzygies (over the symmetric algebra \(S^\ast V\)) naturally generalizes to the exterior algebra \(E:={\textstyle\bigwedge}^{\ast}V\). Let \(P\) be a finitely generated graded \(E\)-module, and consider the \(i\)-th free module \[\bigoplus_{j\in\mathbb{Z}}K^{i,j}(P)\otimes E(-i-j)\] of the minimal graded \(E\)-free resolution of \(M\), where \(K^{i,j}(P)\) is the cohomology group of the Koszul type complex \[\begin{tikzcd} S^{i+1}V\otimes P_{j-1} \ar[r] & S^iV\otimes P_j \ar[r] & S^{i-1}V\otimes P_{j+1} \end{tikzcd}\] at the middle. Let \(F\subseteq\operatorname{Gr}(\mathbb{P}V):=\bigsqcup_{s=0}^{\dim\mathbb{P}V}\operatorname{Gr}(\mathbb{P}^s,\mathbb{P}V)\) be a subvariety of a Grassmannian for \(\mathbb{P}V\). To it we assign a homogeneous ideal \[J(F)=\{w\in E:w|_\Lambda=0\text{ for all }[\Lambda]\in F\}\subseteq E,\] where we denote by \(w|_\Lambda\) the image of \(w\) under the restriction map \({\textstyle\bigwedge}^{\ast}H^0(\mathcal{O}_{\mathbb{P}V}(1))\to{\textstyle\bigwedge}^{\ast}H^0(\mathcal{O}_\Lambda(1))\).

Returning to our central objects we define some subvarieties of \(\operatorname{Gr}(\mathbb{P}V)\) by abuse of notation.

Then with the induced exterior ideals \(J(\sigma_q\tau^kX)\) our theorems generalize in the exterior setting. For example, we state the following.

Also, \(K^{p,q+1}(U,W)\) has a meaning in terms of exterior syzygies. Let \(Y=\mathbb{P}U\times\mathbb{P}W\subseteq\mathbb{P}(U\otimes W)\) be the Segre variety with two factors, and define \[\sigma_q(Y/\mathbb{P}W)=\{\mathbb{P}(U\otimes\overline{W}):\overline{W}\twoheadleftarrow W\mathrm{ is a quotient of dimension }q\}\subset\operatorname{Gr}(\mathbb{P}(U\otimes W)).\]

Accordingly, it can be said that our proofs implicitly pass through the exterior world. In particular, the Green-Lazarsfeld products \(\boxtimes\) and their symmetric-symmetric versions lead to the following decompositions that display hidden interplays between symmetric and exterior bottom syzygies of \(\sigma_q\tau^kY\) for \((2k+2)\)- or \((2k+1)\)-factor Segre varieties \(Y\) with \(q\geq1\) and \(k\geq0\) varying.

1. If \(Y=Y_1\times Y_2:=(\mathbb{P}U_1\times\cdots\times\mathbb{P}U_{2k_1+1})\times(\mathbb{P}W_1\times\cdots\times\mathbb{P}W_{2k_2+1})\subseteq\mathbb{P}(U_1\otimes\cdots\otimes U_{2k_1+1}\otimes W_1\otimes\cdots\otimes W_{2k_2+1})\) has \(2k+2:=2(k_1+k_2)+2\) factors of projective spaces, then \[K_{c,q+1}(I(\sigma_q\tau^kY))=\bigoplus_{a+b=c}K^{a,b+q+1}(J(\sigma_{b+q+1}\tau^{k_1}Y_1))\otimes K^{b,a+q+1}(J(\sigma_{a+q+1}\tau^{k_2}Y_2)).\]

2. If \(Y=Y_1\times Y_2:=(\mathbb{P}U_1\times\cdots\times\mathbb{P}U_{2k_1+2})\times(\mathbb{P}W_1\times\cdots\times\mathbb{P}W_{2k_2+1})\subseteq\mathbb{P}(U_1\otimes\cdots\otimes U_{2k_1+2}\otimes W_1\otimes\cdots\otimes W_{2k_2+1})\) has \(2k+1:=2(k_1+k_2+1)+1\) factors of projective spaces, then \[K^{c,q+1}(J(\sigma_q\tau^kY))=\bigoplus_{a+b=c}K_{a,b+q+1}(I(\sigma_{b+q+1}\tau^{k_1}Y_1))\otimes K^{b,a+q+1}(J(\sigma_{a+q+1}\tau^{k_2}Y_2)).\]

3. If \(Y=Y_1\times Y_2:=(\mathbb{P}U_1\times\cdots\times\mathbb{P}U_{2k_1+2})\times(\mathbb{P}W_{3}\times\cdots\times\mathbb{P}W_{2k_2+2})\subseteq\mathbb{P}(U_1\otimes\cdots\otimes U_{2k_1+2}\otimes W_1\otimes\cdots\otimes W_{2k_2+2})\) has \(2k+2:=2(k_1+k_2+1)+2\) factors of projective spaces, then \[K_{c,q+1}(I(\sigma_q\tau^kY))=\bigoplus_{a+b=c}K_{a,b+q+1}(I(\sigma_{b+q+1}\tau^{k_1}Y_1))\otimes K_{b,a+q+1}(I(\sigma_{a+q+1}\tau^{k_2}Y_2)).\]

We would like to thank Daniele Agostini and Jinhyung Park for valuable discussions and for kindly sharing their manuscript. The author is supported by a KIAS Individual Grant (MG083302) at Korea Institute for Advanced Study.

References↩︎

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1. We have been informed by Daniele Agostini and Jinhyung Park that they carried out the same study independently. See [4].↩︎

2. If \(k=0\), then the lower bound \(\deg L\geq 3g+3q+1\) also works by [4], yielding an improvement when \(q\) is small with respect to \(g\).↩︎