Gorenstein Terminal 3-Fold

• Gorenstein terminal 3-folds are normal projective varieties of dimension three with terminal singularities and a Cartier canonical class.
• They play a central role in the minimal model program by engaging G-invariant Picard groups and enforcing rigid birational contractions.
• Classification leverages explicit constructions such as blow-ups, double covers, and divisors in product spaces to illustrate concrete geometric models.

A Gorenstein terminal 3-fold is a normal, projective three-dimensional variety over a field of characteristic zero which has Gorenstein, terminal singularities. Terminality requires that all discrepancies are strictly positive, while the Gorenstein property (index one case) ensures the canonical class is Cartier. The paper of such 3-folds sits at the center of modern birational algebraic geometry, particularly in the minimal model program (MMP), classification theory, and the geometry of Fano varieties.

1. Structure and Picard Data: G-Fano Condition

Let $X$ be a Fano threefold with at worst Gorenstein terminal singularities. For the rich subclass of “G-Fano threefolds,” consider the additional structure of a finite group $G$ acting by automorphisms (the “geometric case”) or as the Galois group of the base field (the “algebraic case”). The defining property is that the $G$-invariant part of the Weil divisor class group is cyclic: $\operatorname{Cl}(X)^G \cong \mathbb{Z}$ i.e., the only $G$-invariant Weil divisors are integral multiples of $-K_X$ (or its class).

This acts as a symmetry constraint: despite $X$ possibly having Picard number $\rho(X)>1$, only the anticanonical class (or its multiples) are $G$-invariant. This situation occurs in both Galois-invariant and equivariant Fano settings, leading to rigidity in birational geometry.

Key formula: $$\operatorname{Pic}(X)^G \cong \mathbb{Z},\qquad -K_X \text{ generates}$$

2. Birational Geometry: Extremal Contractions and Decomposition

Gorenstein terminal 3-folds $X$ with higher Picard number admit several extremal rays in the Mori cone $\overline{NE}(X)$. Each extremal ray corresponds to a morphism—either a birational contraction (blow-up/blow-down), a conic bundle, or a del Pezzo fibration. The crucial feature under the $G$-Fano condition is that the extremal rays are permuted in $G$-orbits:

• $-K_X$ admits a decomposition as a sum of $G$-related classes, e.g.,

$-K_X = M_1 + M_2 \quad \text{with}\quad T(M_1)=M_2$

for a generator $T\in G$. Such $G$-equivariant decompositions force rigidity on the geometry of $X$, synchronizing the possible contractions and the structure of the ambient space.

The key system of equations is constrained by $G$-symmetry and intersection theory. For any $G$-orbit $\mathcal{O}$ of divisors $F_1, \ldots, F_n$ (e.g., fibers of a del Pezzo fibration), numerical relations such as

$n\cdot (-K_X)^3 = a\,c_3$

arise, where $a$ is integral and $c_3$ is a topological invariant.

3. $G$-Symmetry and Classification Constraints

The group $G$ acts on the Néron-Severi space and must preserve the intersection form and $(-K_X)$. Divisorial data appear in $G$-orbits; extremal contractions and their exceptional loci are $G$-related. This tightens possible configurations, allowing only very specific diagrams of contractions and orbit structures.

Typical types of fibers or exceptional divisors appear as $G$-orbits:

• Del Pezzo fibrations, conic bundles, or birational blowups along $G$-invariant curves. Numerical restrictions on possible $G$-actions emerge from order computations (e.g., via Minkowski's theorem). These further reduce the list of possible threefolds satisfying the $G$-Fano condition.

4. Classification Theorem

For $X$ smooth, with Picard number $p(X)>1$ and $G$ as above preserving both the intersection form and $-K_X$, the classification theorem states:

• $X$ is isomorphic to one of finitely many explicit models, all constructed as either:
  • Complete intersections in products of projective spaces
  • Double covers of homogeneous varieties branched along suitable divisors
  • Blow-ups along well-understood $G$-invariant curves (twisted quartics, conics, elliptics)

Table of principal cases:

Label	Model Description	Invariants/Remarks
(1.2.1)	Divisor of bidegree (2,2) in $\mathbb{P}^2\times\mathbb{P}^2$, or double cover of Segre $V_6$ (branch in $|-K_{V_6}|$)	$p(X)=2$
(1.2.2)	Blow-up of $\mathbb{P}^3$ along a curve of degree $6$, genus $3$	Also described as 3 (1,1) divisors in $\mathbb{P}^3\times\mathbb{P}^3$
(1.2.3)	Blow-up of a Veronese surface in the blow-up of $\mathbb{P}^5$	Intersections in blow-ups of Fano fivefolds

Each case enumerated satisfies the $G$-invariance condition, and any singular $X$ (with terminal Gorenstein singularities) is shown to be smoothable to a member of the list.

5. Numerical and Intersection Theory Constraints

The interplay between the invariant part of the Picard group and the $G$-action yields explicit intersection and symmetry relations:

• $G$-invariance on divisors:

$$M_1 + T(M_1) = -a_1 K_X, \quad M_2 + T(M_2) = -a_2 K_X$$

For most cases, $a_1 = a_2 = 1$.

• The unique $G$-invariant divisor property (in $\operatorname{Pic}(X)\otimes \mathbb{Q}$):

$$\operatorname{Pic}(X)^G \cong \mathbb{Z} \cdot [-K_X]$$

• For a $G$-orbit of fibers $\{F_1, ..., F_n\}$ in a fibration:

$F_i \sim a_i(-K_X), \quad n(-K_X)^3 = a c_3$

• Contractibility constraints: If $X$ contains a contractible plane, it cannot satisfy the $G$-Fano condition.

6. Examples, Rationality, and Further Remarks

• Explicit constructions (involving divisors, double covers, and blowups) realize each classification case.
• Rationality: With the exception of the double cover and (2,2) divisor cases, all classified threefolds are rational.
• Galois cases correspond to taking the Galois group $G$ acting trivially on $X$, reducing to “algebraic” situations.

7. Concluding Framework: Group Actions, Birational Rigidity, and Applications

Gorenstein terminal 3-folds with prescribed $G$-action and Picard number $>1$ serve as a laboratory for the interaction of group symmetries, birational geometry, and classification theory. The $G$-symmetry dramatically reduces allowable models—often to a handful of explicitly described examples—and forces rigid behavior of extremal contractions and divisor orbits. These phenomena underpin foundational rigidity results and equivariant birational geometry, with implications for rationality, moduli, and derived categories.

Key Reference Formulas

• $G$-invariance:

$\operatorname{Cl}(X)^G \cong \mathbb{Z}$

• Anticanonical decomposition with two $G$-related contractions:

$-K_X = M_1 + M_2$

• Intersection constraints:

$n\,(-K_X)^3 = a\,c_3$

The rigid intersection-theoretic and birational structure of Gorenstein terminal 3-folds under strong symmetry constraints thus leads to a concise and constructive classification, as detailed in the referenced work (Prokhorov, 2011).