2. Let \(Y\) be an algebraic variety \(X\) over a field \(\Bbbk\) and let \(G\) be a group. Following works of Yu. I. Manin [1] we say that \(X\) is a \(G\)-variety if the group \(G\) acts on \(\bar X:=X\otimes_\Bbbk \bar \Bbbk\), where \(\bar \Bbbk\) is the algebraic closure of \(\Bbbk\). Moreover, we assume that \(X\), \(G\) and \(\Bbbk\) satisfy one of the following two conditions.

(a) Algebraic case. \(G\) is the Galois group of \(\bar \Bbbk\) over \(\Bbbk\) acting on \(\bar X\) through the second factor. The action of \(G\) on \(X\) is trivial.

(b) Geometric case. The field \(\Bbbk\) is algebraically closed, \(G\) is a finite group, and the action of \(G\) on \(X\) is given by a homomorphism \(G\to \operatorname{Aut}_{\Bbbk}(X)\).

A \(G\)-morphism (resp. rational \(G\)-map) of \(G\)-varieties is a \(\Bbbk\)-morphism (resp. \(\Bbbk\)-rational map) commuting with the action of \(G\) in the geometric case. A projective \(G\)-morphism \(f: X\to Z\) of normal \(G\)-varieties is called \(G\)-Mori fiber space if \(X\) has at worst terminal \(G\mathbb{Q}\)-factorial singularities (see 12), \(f_*\mathscr{O}_X=\mathscr{O}_Z\), the relative invariant Picard group \(\operatorname{Pic}(X/Z)^G\) is of rank \(1\), and the anticanonical divisor \(-K_X\) is \(f\)-ample. In the particular case where \(\dim Z=0\), \(X\) is called a \(G\mathbb{Q}\)-Fano variety. If furthermore the canonical divisor is Cartier, then we say that \(X\) is \(G\)-Fano variety.

Throughout this paper we assume that the ground field has characteristic \(0\). The following is an easy consequence of the Minimal Model Program [2] (cf. [3]).

Birational classification of \(G\)-surfaces is developed very well [1], [4]. In this and subsequent papers we consider \(G\)-Mori fiber spaces \(X\to Z\), where \(\dim X=3\) and \(Z\) is a point, i.e. the case of \(G\mathbb{Q}\)-Fano threefolds.

3. Let \(X\) be a \(G\)-Fano threefold. It is well-known that \(\operatorname{Pic}(X)\) is a finitely generated torsion free abelian group (see, e.g. [5]). Consider the following composed object: \[\operatorname{V}(X)=\bigl(\operatorname{Cl}(X),\operatorname{Pic}(X),K_X,(\phantom{v},\phantom{v},\phantom{v})\bigr),\] where \(\operatorname{Pic}(X)\) is regarded as a sublattice of \(\operatorname{Cl}(X)\), \(K_X\in \operatorname{Pic}(X)\) is the canonical class of \(X\), and \((\phantom{v},\phantom{v},\phantom{v})\) is the intersection form \(\operatorname{Pic}(X)\times \operatorname{Pic}(X)\times\operatorname{Cl}(X)\to \mathbb{Z}\). Since the singularities of \(X\) are isolated cDV [6], \(\operatorname{Pic}(X)\) is a primitive sublattice in \(\operatorname{Cl}(X)\), i.e. the quotient \(\operatorname{Cl}(X)/\operatorname{Pic}(X)\) is torsion free [7]. Moreover, since \(\rho(X)^G=1\), we have \[\text{\operatorname{Cl}(X)^G is a subgroup of rank 1 containing K_X.}\]

4. In this paper we give a classification of one class of Gorenstein \(G\)-Fano threefolds. More precisely, we consider Fano threefolds such that \(-K_X=2S\) for some ample Cartier divisor \(S\). Then \(X\) is called a del Pezzo threefold (see 14). Smooth del Pezzo threefolds were classified by Iskovskikh [8], see also [9], [5]. Singular ones were discussed from different points of view in many works [10], [11], [12], [13], [14]. We are interested basically in group actions on terminal del Pezzo threefolds \(X\) and the structure of the lattice \(\operatorname{Cl}(X)\).

5. Let \(S\) be a smooth del Pezzo surface of degree \(d:=K_S^2\). Then we have \(\operatorname{Pic}(S) = \mathbb{Z}^{10-d}\). Define \[\Delta:=\{\alpha \in \operatorname{Pic}(S) \mid \alpha^2=-2,\quad \alpha\cdot K_S=0\}.\] Then \(\Delta\) is a root system in \((K_S)^{\perp}\otimes \mathbb{R}\). Depending on \(d\), \(\Delta\) is of the following type ([15]):

where \(8'\) (resp. \(8''\)) corresponds to \(\mathbb{P}^1\times\mathbb{P}^1\) (resp. Hirzebruch surface \(\mathbb{F}_1\)).

6. Now let \(X\) be a del Pezzo threefold. Let \(S\in |{-}\frac{1}{2}K_X|\) be a smooth member [11] and let \(\iota: S \hookrightarrow X\) be the natural embedding. Then \(S\) is a del Pezzo surface of degree \(d=-\frac{1}{8} K_X^3\). It is easy to show that the restriction map \(\iota^*: \operatorname{Cl}(X)\to \operatorname{Pic}(S)\) is injective and its cokernel is torsion free (see 27). Define the following subsets in \(\Delta\subset\operatorname{Pic}(S)\): \[\begin{array}{lll} \Delta'&:=&\{\alpha \in \operatorname{Pic}(S) \mid\alpha^2=-2,\;\alpha\cdot K_S= \alpha\cdot \iota^*\operatorname{Cl}(X)=0\}, \\[5pt] \Delta''&:=&\{\alpha \in \iota^*\operatorname{Cl}(X) \mid\alpha^2=-2,\;\alpha\cdot K_S=0 \}. \end{array}\] In other words, \[\Delta'=\Delta\cap (\iota^*\operatorname{Cl}(X))^{\perp}, \qquad \Delta''=\Delta\cap \iota^*\operatorname{Cl}(X).\] If \(\Delta'\) (resp. \(\Delta''\)) is non-empty, then it is a root subsystem in \(\Delta\). Assume that \(X\) is a \(G\)-variety. Then the group \(G\) naturally acts on \(\iota^*\operatorname{Cl}(X)\) and \(\Delta''\) preserving the class of \(K_S\) and the intersection form. Our classification of \(G\)-del Pezzo threefolds is by types of root systems \(\Delta'\) and \(\Delta''\).

8. Remark. For \(\operatorname{d}(X)\le 2\) any del Pezzo threefold automatically has \(G\)-structure (see Remark 16). So, in this case, [class-d611-factorial] – [class-maximal-rk-d612] is a complete list of del Pezzo threefolds with \(\operatorname{d}(X)\le 2\).

9. Remark. Singular three-dimensional cubics (without group action) whose singularities are only nodes and their small resolutions were classified in [16]. There is the following correspondence between our list and the classification in [16]: 1 \(\longleftrightarrow\) J15, [class-cubic-subSegre] \(\longleftrightarrow\) J14, [class-cubic-empty-delta] \(\longleftrightarrow\) J11, [class-cubic-P1-bundle]\(\longleftrightarrow\) J9, [class-cubic-factorial]\(\longleftrightarrow\) J1–J5.

10. Notation. We work over an algebraically closed field of characteristic \(0\). Throughout this paper \(X\) denotes a del Pezzo threefold with at worst terminal Gorenstein singularities. Thus we can write \(-K_X=2S\), where \(S=S_X\) is a ample Cartier divisor of \(S\) defined up to linear equivalence. Everywhere below we use the following notation:

• \(\rho=\rho(X):=\operatorname{rk}\operatorname{Pic}(X)\);

• \(\operatorname{Cl}(X)\) is the Weil divisor class group;

• \(\operatorname{r}=\operatorname{r}(X):=\operatorname{rk}\operatorname{Cl}(X)\);

• \(\operatorname{d}=\operatorname{d}(X):=S^3=-K_X^3/8\), the degree of \(X\);

• \(\operatorname{p}=\operatorname{p}(X)\) is the number of planes on \(X\);

• \(\operatorname{s}=\operatorname{s}(X)\) is the number of singular points of \(X\) under an additional assumption that \(X\) has at worst nodes;

• \(V_6\subset \mathbb{P}^7\) is a smooth del Pezzo threefold with \(\operatorname{d}(X)=6\) and \(\rho=2\), see Theorem 15;

• \(V_5\subset \mathbb{P}^6\) is a smooth del Pezzo threefold of degree \(5\) (see [5]);

• \(V_d\), for \(d=1,\dots, 4\), is a del Pezzo threefold of degree \(d\) with terminal factorial singularities (see Theorem 15).

12. Terminal singularities (see [6]). Let \((X, P)\) be a germ of a three-dimensional terminal singularity. Then \((X, P)\) is isolated, i.e, \(\operatorname{Sing}(X)=\{P\}\). If additionally \((X, P)\) is Gorenstein, then \((X, P)\) is analytically isomorphic to a hypersurface singularity of multiplicity \(2\).

Let \(X\) be a threefold with Gorenstein terminal singularities. Then any Weil \(\mathbb{Q}\)-Cartier divisor is Cartier (see e.g. [7]). Equivalently, \(\operatorname{Pic}(X)\) is a primitive sublattice in \(\operatorname{Cl}(X)\).

Let \(X\) be a \(G\)-variety. We say that \(X\) has only \(G\mathbb{Q}\)-factorial singularities if any \(G\)-invariant Weil divisor is \(\mathbb{Q}\)-Cartier.

14. Definition. Let \(X\) be a projective variety \(X\) with at worst terminal Gorenstein singularities.¹ We say that \(X\) is a del Pezzo threefold (resp. weak del Pezzo threefold) if its anti-canonical class \(-K_X\) is divisible by \(2\) and is ample (resp. nef and big).

23. Definition. Let \(X\) be a weak del Pezzo threefold and let \(S=-\frac{1}{2}K_X\). An irreducible surface \(\Pi\subset X\) is called a plane if \(S^2\cdot \Pi=1\) and, in case \(\operatorname{d}(X)=1\), the base point of \(|S|\) does not lie on \(\Pi\).

32. Remark. Let \(X\) be a primitive del Pezzo threefold with \(\operatorname{r}(X)=3\) and \(\operatorname{d}(X)=2\) or \(4\). Let \(\xi : \hat{X}\to X\) be a \(\mathbb{Q}\)-factorialization. Then \(\hat{X}\simeq \mathbb{P}({\mathscr{E}})\), where \({\mathscr{E}}\) is a stable rank two vector bundle on \(Z=\mathbb{P}^1\times \mathbb{P}^1\) with \(c_1({\mathscr{E}})=0\), \(c_2({\mathscr{E}})=6-\operatorname{d}(X)\).

35. Let \(X\) be a del Pezzo threefold with \(\operatorname{r}(X)=2\). There exists the following diagram: \[\xymatrix{ &\hat{X}\ar[dl]_{f}\ar[dr]_{\xi}\ar@{-->}[rr] &&\hat{X}^+\ar[dl]^{\xi^{+}}\ar[dr]^{f^+} \\ Z&&X&&Z^+ }\] where \(\xi\), \(\xi^+\) are small \(\mathbb{Q}\)-factorializations, \(\hat{X} \dashrightarrow \hat{X}^+\) is a flop, and \(f\), \(f^+\) are \(K\)-negative extremal contractions. We may assume that \(\dim Z\ge \dim Z^+\). Let \(S=-\frac{1}{2} K_X\) and let \(\hat{S}=h^*S\). Let \(M\) (resp. \(M^+\)) be the ample generator of \(\operatorname{Pic}(Z)\) (resp. \(\operatorname{Pic}(Z^+)\)). Put \(L:=f^*M\) and \(L^+:=f^{{+}*}M^+\). Let \(L'\) be the proper transform of \(L^+\) on \(\hat{X}\). If \(f\) is birational, then \(E\subset \hat{X}\) denotes the \(f\)-exceptional divisor. Similarly, if \(f^+\) is birational, then \(E'\subset \hat{X}\) is the proper transform of \(f^+\)-exceptional divisor.

39. By [14] all the cases in the table do occur². Below we give explicit examples of some del Pezzo threefolds with \(\operatorname{r}(X)=2\).

Case [rkCl612-P1-bundle-degree613].↩︎

\(X=X_3\subset \mathbb{P}^4\) is given by an equation of the form \[(x_1x_4-x_2x_3)\ell_1 +(x_2^2-x_1x_3)\ell_2 +(x_3^2-x_2x_4)\ell_3=0,\] where \(\ell_i(x_1,\dots, x_5)\) are linear forms.

Case [rkCl612-degree615]↩︎

(cf. 54 and 68.) Let \(Y\) be the blowup of \(\mathbb{P}^1\times \mathbb{P}^2\) along a smooth curve \(C\) of bidegree \((2,1)\). Then \(Y\) is a Fano threefold with \(-K_Y^3=38\) and \(\rho(Y)=3\) [24]. Let \(S\subset \mathbb{P}^1\times \mathbb{P}^2\) be a (unique) effective divisor of bidegree \((0,1)\) containing \(C\) and let \(\tilde{S}\) be the proper transform of \(S\) on \(Y\). Then \(\tilde{S}\simeq S\simeq \mathbb{P}^1\times \mathbb{P}^1\) and \(\mathscr{O}_{\tilde{S}}(\tilde{S})\) is of type \((-1,-1)\). Therefore, there exists a contraction \(\varphi: Y\to X\), where \(\varphi(\tilde{S})\) is a node. Here \(X\) is a quintic del Pezzo threefold as in [rkCl612-degree615].

Case [rkCl612-deg2-quadric].↩︎

\(X\subset \mathbb{P}(1^4,2)\) is given by the equation \[x_5^2=(x_1x_2-x_3x_4)^2+ (x_1x_2-x_3x_4) q_1(x_1,\dots,x_4)+q_2(x_1,\dots,x_4)^2,\] where \(q_1\) and \(q_2\) are general quadratic forms.

Case [rkCl612-deg4-quadric].↩︎

\(X\subset \mathbb{P}^5\) is given by the equations \[x_1x_2+x_3x_4+x_5^2+x_6l_1(x_1,\dots x_6)= x_1x_3+x_6l_2(x_1,\dots x_6)=0,\] where \(l_i\) are linear forms. It is easy to see that \(X\) contains two singular quadrics given by \(x_6=x_1=x_3x_4+x_5^2=0\) and \(x_6=x_3=x_1x_2+x_5^2=0\). They generate two pencils. Hence \(X\) is of type [rkCl612-deg4-quadric]. For a general choice of \(l_i\) the variety \(X\) has exactly one node.

Case [rkCl612-deg4-birational].↩︎

\(X\subset \mathbb{P}^5\) is given by the equations \[x_3x_4-x_5^2+x_6l_1(x_1,\dots x_6)= x_1x_4-x_2x_5+x_6 l_2(x_1,\dots x_6)=0,\] where \(l_i\) are general linear forms. Then \(X\) contains the plane \(\Pi:=\{x_4=x_5=x_6=0\}\). The singular locus of \(X\) consists of three nodes contained in \(\Pi\) and given there by \(x_3=l_1=0\) and \(x_2=x_3l_2-x_1l_1=0\).

Case [rkCl612-cubic-plane].↩︎

Let \(X\subset \mathbb{P}^4\) be given by the following equation: \[x_1u(x_1,x_2,x_3,x_4,x_5)+x_2v(x_1,x_2,x_3,x_4,x_5)=0,\] where \(u\) and \(v\) are quadratic forms. This cubic contains the plane \(\Pi:=\{x_1=x_2=0\}\) and, for general \(u\) and \(v\), the singular locus consists of four nodes. The projection from \(\Pi\) gives us a quadric bundle structure on \(\hat{X}\) (which is the blowup of \(\Pi\)).

Case [rkCl612-deg1-birational].↩︎

\(X\subset \mathbb{P}(1^3,2,3)\) is given by the equation \[x_5^2=x_4^3+ x_4^2\phi_2 + x_4\phi_4 + \phi_3^2,\] where \(\phi_i(x_1,x_2,x_3)\) are general homogeneous forms of degree \(i\).

Case 2.↩︎

\(X\subset \mathbb{P}(1^4,2)\) is given by the equation \[x_5^2=x_1\phi_3(x_1,\dots,x_4)+q(x_1,\dots,x_4)^2,\] where \(\phi_3\) and \(q\) are general homogeneous forms of degree \(3\) and \(2\), respectively.

40. Let \(X\) be a del Pezzo threefold of degree \(d=\operatorname{d}(X)\). In this section we study the image of the restriction map \(\iota^*: \operatorname{Cl}(X) \to \operatorname{Pic}(S)\), where \(S\in |{-}\frac{1}{2}K_X|\) is a smooth member contained in the smooth locus of \(X\) and \(\iota: S\hookrightarrow X\) is an embedding. Define \(\Delta\) and \(\Delta'\) as in 6. If \(X\) is imprimitive, we apply construction ?? with all corresponding notation. In the primitive case, to unify notation, we put \(\sigma=\operatorname{id}\).

Note that \(S\) does not pass through singular points of \(X\). Thus we may identify \(S\) and \(\hat{S}=\xi^{-1}(S)\). Let \(\bar S:= \sigma(S)\). Then \(\bar S\) is a smooth del Pezzo surface, \(\bar S\in |{-}\frac{1}{2} K_{\bar X}|\) and \(\sigma_S: S\to \bar S\) is a blowup of \(\operatorname{r}(X)-\operatorname{r}(\bar X)\) distinct points. Define \(\bar \Delta\) and \(\bar \Delta'\) for \(\bar S\) as in 6.

53. Sextic del Pezzo threefold.Let \(X\subset \mathbb{P}^2\times \mathbb{P}^2\) be given by the equation \(x_1y_1+x_2y_2=0\). Then \(X\) is a del Pezzo threefold with \(\operatorname{d}(X)=6\) and \(\operatorname{r}(X)=3\). The singular locus consists of one node.

54. Quintic del Pezzo threefold (cf. [28]).Let \(X\subset \operatorname{Gr}(2,5)\) be an intersection of three general Schubert subvarieties of codimension one. Then \(X\) is a del Pezzo threefold with \(\operatorname{d}(X)=5\) and \(\operatorname{r}(X)=4\). The singular locus consists of three nodes.

55. Quartic del Pezzo threefold. Let \(X\subset\mathbb{P}^5\) be an intersection of two quadrics having \(6\) isolated singular points. Then in some coordinate system \(X\) can be given by the equations \[\label{equation-maximal-deg614} x_1^2-x_2^2=x_3^2-x_4^2=x_5^2-x_6^2.\tag{2}\] In [27] this variety is called the tetrahedral quartic threefold. By Corollary 93 \(\operatorname{r}(X)=5\). The variety \(X\) contains \(8\) planes \[\Pi_{\epsilon_1,\epsilon_2,\epsilon_3}=\{ x_1+ \epsilon_1 x_2=x_3 +\epsilon_2 x_4=x_5+\epsilon_3 x_6=0\},\] where \(\epsilon_i=\pm 1\). Clearly, \[\dim \Pi_{\epsilon_1,\epsilon_2,\epsilon_3}\cap \Pi_{\epsilon_1',\epsilon_2',\epsilon_3'}= -1+\frac{1}{2}\sum |\epsilon_i+\epsilon_i'|.\] Therefore, for each plane \(\Pi=\Pi_{\epsilon_1,\epsilon_2,\epsilon_3}\) there is exactly \(3\) planes \(\Pi'\) such that \(\Pi\cap \Pi'\) is a point and exactly \(3\) planes \(\Pi'\) such that \(\Pi\cap \Pi'\) is a line. Note that there are two \(4\)-tuples of planes such that planes in each tuple meet each other only by subsets of dimension \(\le 0\): \[\{\Pi_{+++},\, \Pi_{+--},\, \Pi_{-+-},\, \Pi_{--+}\},\quad \{\Pi_{---},\, \Pi_{-++},\, \Pi_{+-+},\, \Pi_{++-}\}.\] The involution \[\tau: (x_1,x_2,x_3,x_4,x_5,x_6) \longmapsto (x_1,-x_2,x_3,-x_4,x_5,-x_6)\] interchanges these \(4\)-tuples. Hence \(\tau\) induces a birational (cubo-cubic) involution on \(\mathbb{P}^3\). In [29] it is denoted by \(T_{\mathrm{tet}}\). Note however that \(\operatorname{Cl}(X)^{\tau}\not \simeq \mathbb{Z}\), i.e. \(X\) is not \(\tau\)-minimal. \(X\) is minimal with respect to the whole automorphism group.

56. Segre cubic. If \(\operatorname{d}(X)=3\), then \(X\) is can be given by \[\label{equation-Segre-cubic} X=X_3^{\mathrm s}=\left\{\sum_{i=1}^6 x_i= \sum_{i=1}^6 x_i^3=0\right\}\subset \mathbb{P}^4\subset \mathbb{P}^5.\tag{3}\] This cubic satisfies many remarkable properties (see [27]) and is called the Segre cubic. For example, any cubic hypersurface in \(\mathbb{P}^4\) has at most ten isolated singular points, this bound is sharp and achieved exactly for the Segre cubic (up to projective isomorphism). The symmetric group \(\mathfrak{S}_6\) acts on \(X_3^{\mathrm s}\) in the standard way. Moreover, by Corollaries 50 and 51 we see that \(\operatorname{Aut}(X_3^{\mathrm s})=\mathfrak{S}_6\), so the natural map \(\operatorname{Aut}(X_3^{\mathrm s})\to \operatorname{W}(\Delta'')\) is an isomorphism.

58. Quartic double solid. Let \(X\) be a del Pezzo threefold of degree \(2\). Let \(\phi: X \to \mathbb{P}^3\) be the half-anticanonical map. Then \(\phi\) is a double cover whose branch locus \(B\subset \mathbb{P}^3\) is a quartic having \(16\) singular points. It is well-known that such a quartic must be a Kummer surface, so the singularities of \(B\) and \(X\) are at worst nodes [30], [31] (see also [32]). The singular points of \(X\) correspond to \(15\) lines \(L_{ij}\) passing through pairs of points \(P_i\), \(P_j\) and one twisted cubic passing through all points \(P_1,\dots,P_6\). The threefold \(X\) contains \(32\) planes [27]. For each such a plane \(\Pi\) the image \(\pi(\Pi)\) is a plane touching \(B\) along a conic.

60. Double Veronese cone. Recall that \(X\simeq X_6\subset \mathbb{P}(1^3,2,3)\). The projection from \((0,0,0,0,1)\) indices a double cover \(X\to \mathbb{P}(1^3,2)\) with branch divisor \(B=B_6\subset \mathbb{P}(1^3,2)\). Assume for simplicity that the singularities of \(X\) are at worst nodes. Then \(B\) is a surface having exactly \(28\) points of type \(A_1\). Conversely if \(B\subset \mathbb{P}(1^3,2)\) is a surface of degree 6 whose singularities are exactly \(28\) points of type \(A_1\), then the double cover of \(\mathbb{P}(1^3,2)\) branched at \(B\) is a del Pezzo threefold with \(\operatorname{d}(X)=1\) and \(\operatorname{r}(X)=8\). We refer to [33] for more detailed treatment and more references.

68. Quintic del Pezzo threefold (cf. [28]).Let \(X\subset \operatorname{Gr}(2,5)\) be an intersection of two general Schubert subvarieties of codimension one and one general hyperplane section. Then \(X\) is a del Pezzo threefold with \(\operatorname{d}(X)=5\) and \(\operatorname{r}(X)=3\). The singular locus consists of two nodes.

69. Quartic del Pezzo threefold.Let \(X\subset \mathbb{P}^5\) be given by the equations \[x_1^2+x_1x_3+x_2x_5=x_1x_3+x_3^2+ x_4x_6=0.\] Then \(X\) is a del Pezzo threefold of degree \(4\) containing exactly \(5\) nodes. By Corollary 93 \(\operatorname{r}(X)\ge 4\). On the other hand, \(X\) is not of type 55 because \(\operatorname{s}(X)<6\). Hence \(\operatorname{r}(X)=4\).

70. Cubic hypersurface. Let \(X\subset \mathbb{P}^4\) be given by the equation \[x_1x_2\ell(x_1\dots,x_5)+(x_3x_4+x_1x_2)x_5=0,\] where \(\ell\) is a general linear form. Then \(X\) is a cubic del Pezzo threefold with \(\operatorname{s}(X)=9\), \(\operatorname{r}(X)=5\), and \(\operatorname{p}(X)=9\) (cf. [16]).

71. Quartic double solid. Let \(Y\) be a hypersurface in \(\mathbb{P}^4\) given by \(\{s_1=4s_4-s_2^2=0\}\subset \mathbb{P}^5\), where \(s_k=\sum x_i^k\). This famous hypersurface is called Igusa quartic. The singular locus of \(Y\) consists of \(15\) lines. Consider a general hyperplane section \(B:=Y\cap \mathbb{P}^3\). Then \(B\) is a quartic having \(15\) nodes (cf. [32]). Let \(X\to \mathbb{P}^3\) be a double cover with branch divisor \(B\). Then \(X\) is a del Pezzo threefold of degree \(2\) with \(\operatorname{s}(X)=15\) and \(\operatorname{r}(X)=6\).

72. In this section we prove Theorem 7. We use notation of 40. Furthermore we assume that \(X\) is a \(G\)-del Pezzo threefold. Thus \(\operatorname{Cl}(X)^G\simeq \mathbb{Z}\). By Theorem 37 we may assume that \(\operatorname{r}(X)\ge 3\).

75. From now on we assume that \(\operatorname{d}(X)\le 4\). By Theorem 31 we may assume that \(X\) is imprimitive. Let \(S\in |{-}\frac{1}{2} K_X|\) be a general member. Let \(n:=\operatorname{rk}\Delta=9-\operatorname{d}(X)\).

78. Now we assume that \(\Delta''\neq \emptyset\). Then \(\Delta''\) is a \(G\)-invariant root subsystem in \(\Delta\). By the results of §7 and §8 we may assume that \(\operatorname{r}(X)\le 7-\operatorname{d}(X)\). Further, by Lemma 74 \(\operatorname{d}(X)\le 3\).

82. Consider the case \(\operatorname{d}(X)=3\). There are only the following possibilities:

83. Finally we consider cases \(\operatorname{d}(X)\le 2\). According to Remark 16 any del Pezzo threefold with \(\operatorname{d}(X)\le 2\) is automatically \(G\)-del Pezzo. Thus all the possibilities for \(\bar X\) with \(2\le \operatorname{d}(\bar X)\le 5\) and \(\operatorname{r}(\bar X)\le 2\) do occur (recall that \(3\le \operatorname{r}(X)\le 7-\operatorname{d}(X)\)):

• \(\bar X=V_5\) \(\Longrightarrow\) \(\operatorname{d}(X)\le 2\), \(\Delta'\simeq A_4\);

• \(\bar X=V_4\) \(\Longrightarrow\) \(\operatorname{d}(X)\le 2\), \(\Delta'\simeq D_5\);

• \(\bar X=V_3\) \(\Longrightarrow\) \(\operatorname{d}(X)\le 2\), \(\Delta'\simeq E_6\);

• \(\bar X\) is of type [rkCl612-P1-bundle-degree612] \(\Longrightarrow\) \(\operatorname{d}(X)=1\), \(\Delta'\simeq A_6\);

• \(\bar X\) is of type [rkCl612-P1-bundle-degree613] \(\Longrightarrow\) \(\operatorname{d}(X)\le 2\), \(\Delta'\simeq A_5\);

• \(\bar X\) is of type [rkCl612-degree615] \(\Longrightarrow\) \(\operatorname{d}(X)\le 2\), \(\Delta'\simeq A_3\);

• \(\bar X\) is of type [rkCl612-deg2-quadric] \(\Longrightarrow\) \(\operatorname{d}(X)=1\), \(\Delta'\simeq D_6\);

• \(\bar X\) is of type [rkCl612-deg4-quadric] \(\Longrightarrow\) \(\operatorname{d}(X)\le 2\), \(\Delta'\simeq D_4\).

The number of planes can be found by using Lemma 21 and direct computations.

85. Definition. Let \(V\ni P\) be a threefold terminal Gorenstein (=isolated cDV) singularity. We say that \(V\ni P\) is r-nondegenerate (resolution nondegenerate) if there is a resolution \[\sigma:V_m \overset{\sigma_m} {\longrightarrow} \cdots \overset{\sigma_2} {\longrightarrow} V_1\overset{\sigma_1} {\longrightarrow} V=V_0,\] where each \(\sigma_i\) is a blowup of a singular point \(P_{i-1}\in V_{i-1}\). Such a resolution \(\sigma\) is called standard. In this situation, all varieties \(V_i\) also have only isolated cDV singularities. If furthermore each \(\sigma_i\)-exceptional divisor \(E_i\subset V_i\) is irreducible, then we say that \(V\ni P\) is rs-nondegenerate (strongly resolution nondegenerate).

Denote \(\lambda(V,P):=m\) and let \(\nu(V,P)\) be the number of \(\sigma\)-exceptional divisors. Thus \(\lambda(V,P)\le \nu(V,P)\) and the equality holds if and only if \(V\ni P\) is rs-nondegenerate.

86. Remark. Let \(V\ni P\) be a threefold terminal Gorenstein point and let \(\sigma_1: V_1\to V\) be the blowup of \(P\). Since \(V\ni P\) is a hypersurface singularity, we have an (analytic) embedding \(V_1\subset \tilde{\mathbb{C}}^4\), where \(\tilde{\sigma}_1:\tilde{\mathbb{C}}^4\to \mathbb{C}^4\) is the blowup of the origin. Let \(D:=\tilde{\sigma}_1^{-1}(P)\) be the exceptional divisor. Then \(D\simeq \mathbb{P}^3\). Since \(V\ni P\) is a singularity of multiplicity \(2\), the intersection \(D\cap V_1\) is a quadric in \(\mathbb{P}^3\) (possibly reducible or non-reduced). If \(D\cap V_1\) irreducible, then \(V_1\) is either smooth or has (a unique) terminal singularity. Moreover, the above arguments show that \(2\lambda(V,P)\ge \nu(V,P)\).

89. Let \(W\) be a smooth projective fourfold and let \(V\subset W\) be an effective divisor. Define \[\beta(W,V):=c_3(W)\cdot V-c_2(W)\cdot V^2+c_1( W)\cdot V^3- V^4.\] If \(V\) is smooth then \(\beta(W,V)\) coincides with \(\deg c_3(V)=\operatorname{Eu}(V)\), the topological Euler number.

94. Give a complete birational classification of del Pezzo threefolds over \(\mathbb{C}\). Non-trivial cases only are factorial del Pezzo threefolds of degree \(\le 3\). All other cases can be reduced to the above ones by using construction ?? (or birationally trivial). It is well-known that a three-dimensional cubic hypersurface with at worst cDV singularities is rational if and only if it is singular [34]. A general smooth (and, in some cases, factorial) del Pezzo threefold of degree \(\le 2\) is not rational [35], [36], [37].

96. Give a complete birational classification of del Pezzo threefolds over algebraically non-closed fields. Here is one example.

97. Describe automorphism groups of del Pezzo threefolds over an algebraically closed fields. Which of them are birationally rigid (cf. [39], [40])? These questions are very useful for applications to the classification of finite subgroups of Cremona group \(\operatorname{Cr}_3(\Bbbk)\) [3], [17].