Projective Factorial Variety

• Projective factorial varieties are normal projective varieties defined by the condition that every Weil divisor is Cartier, indicating a defect-free divisor class structure.
• They are characterized by cohomological invariants, where the Q-factoriality defect is computed as the difference between top and low-degree cohomology dimensions, reflecting singularity properties.
• For toric and LCI examples, combinatorial algorithms and cyclic covering techniques classify projectivity and factoriality, underpinning advances in the minimal model program.

A projective factorial variety is a normal projective algebraic variety exhibiting a precise relationship between its divisor class groups and the structure of its singularities. The factoriality of such a variety concerns the distinction between Cartier and Weil divisors, with $\mathbf{Q}$-factoriality capturing whether every Weil divisor is Cartier up to some multiple. The study of factoriality and related invariants, such as the $\mathbf{Q}$-factoriality defect, is central to higher-dimensional birational geometry and the minimal model program, particularly in relation to local complete intersection (LCI) and Cohen–Macaulay singularities. Recent advances provide both cohomological and combinatorial criteria, as well as algorithmic tools for toric cases, for the detection of projectivity and factoriality.

1. Definitions: Factoriality, $\mathbf{Q}$‑Factoriality, and the Defect Invariant

Let $X$ be a normal projective variety over $\mathbb{C}$ of dimension $n \geq 2$. Denote by Weil$(X)$ the free abelian group generated by irreducible Weil divisors on $X$, and by Cartier$(X)$ the subgroup of Weil$(X)$ consisting of Cartier divisors (those that are Zariski-locally principal).

The quotient group Weil$(X) /$ Cartier$(X)$ is finitely generated; its rank is called the $\mathbf{Q}$-factoriality defect:

$$\sigma(X) := \operatorname{rank}_{\mathbb{Z}}( \mathrm{Weil}(X) / \mathrm{Cartier}(X) ).$$

A normal variety is called factorial if every Weil divisor is Cartier (i.e., $\sigma(X) = 0$), and $\mathbf{Q}$-factorial if every Weil divisor is $\mathbf{Q}$-Cartier (i.e., Weil$(X)/$Cartier$(X)$ is torsion).

There is an exact sequence connecting these notions:

$$0 \longrightarrow \mathrm{Weil}(X)/\mathrm{Cartier}(X) \longrightarrow \mathrm{Cl}(X) \longrightarrow \mathrm{Pic}(X) \longrightarrow 0,$$

where Cl$(X)$ is the class group and Pic$(X)$ the Picard group.

2. Cohomological Formula and 2‑Semi‑Rationality

A foundational result is a topological formula that expresses $\sigma(X)$ in terms of the rational cohomology of $X$ provided the singularities are sufficiently mild. Let $\pi: \widetilde X \to X$ be a desingularization. The variety $X$ is said to have 2-semi-rational singularities if

$$R^k \pi_* \mathcal{O}_{\widetilde X} = 0 \quad \text{for} \quad k=1,2.$$

This condition is weaker than rational singularity and is satisfied, for example, by Cohen–Macaulay varieties with codim$_X \text{Sing} X \geq 4$. For such $X$ (where $n := \dim X > 2$),

$\sigma(X) = h^{2n-2}(X) - h^2(X),$

with $h^k(X) := \dim H^k(X, \mathbb{Q})$.

The proof exploits perverse sheaf methods, the decomposition of cohomology via a resolution, and the analysis of the associated mixed Hodge modules and their weight filtrations. The formula generalizes results of Namikawa–Steenbrink for $n=3$ and isolated hypersurface singularities to arbitrary dimension and more general singular sets (Jung et al., 19 Jan 2026).

3. Projectivity and Factoriality for Local Complete Intersections

For $X$ a local complete intersection (LCI) whose singular locus has codimension at least three, $\mathbf{Q}$-factoriality implies factoriality. Explicitly, every effective $\mathbf{Q}$-Cartier divisor is already Cartier:

• If $X$ is $\mathbf{Q}$-factorial and LCI with codim Sing $X\geq 3$, then $X$ is factorial.

The proof utilizes cyclic covering techniques at a singular point. It is shown that every such cover is topologically trivial over a punctured neighborhood, using arguments from the theory of perverse sheaves and the vanishing cycle formalism, extending classical results in dimension three and for isolated hypersurface singularities.

Moreover, Grothendieck’s para-factoriality theorem in the projective context is recovered: An LCI $X$ with codim Sing $X\geq 4$ is factorial, as the topological formula then yields $\sigma(X)=0$ (Jung et al., 19 Jan 2026).

4. The Case of Cohen–Macaulay Varieties and $\mathbf{Q}$‑Factoriality

If $X$ is projective, Cohen–Macaulay, and $\operatorname{codim}_X \mathrm{Sing}\, X \geq 4$, then $R^k\pi_*{\mathcal{O}_{\widetilde{X}}}=0$ for $k=1,2$, so the defect formula again gives $\sigma(X)=0$ up to torsion. Consequently, $X$ is $\mathbf{Q}$-factorial, but need not be factorial. For example, for the affine cone $X$ over the Veronese embedding $Y\subset \mathbb{P}^{n-1}$, one finds $Cl(X)\cong \mathbb{Z}/m$, $Pic=0$, so $X$ is $\mathbf{Q}$-factorial but not factorial (Jung et al., 19 Jan 2026).

This shows that for Cohen–Macaulay/projective varieties with high codimension singular locus, $\mathbf{Q}$-factoriality and factoriality diverge, in contrast to the LCI/codim$\geq 3$ case.

5. Toric Projective $\mathbf{Q}$‑Factorial Varieties: Combinatorial and Algorithmic Aspects

A projective toric variety $X(\Sigma)$ associated to a fan $\Sigma$ is $\mathbf{Q}$-factorial if and only if every maximal cone is simplicial. Let $$V = \{v_1, \ldots, v_m\}\subset N \simeq \mathbb{Z}^n$$ be primitive ray generators. The number of rays $m$ satisfies $m = n + r$, where $r$ is the Picard number.

Projectivity is characterized by the existence of a strictly convex support function $\psi$ linear on cones of $\Sigma$ and $\mathbb{Z}$-valued on $V$. The nef cone is

$$\mathrm{Nef}(X) = \bigcap_{\sigma \in \Sigma(n)} \mathrm{Cone}(\{D_i \mid v_i \notin \sigma\}),$$

and $X(\Sigma)$ is projective if and only if $\mathrm{Nef}(X)$ is full dimensional ($\dim = r$).

Algorithmically, two procedures allow for the classification of all complete and simplicial fans with a given $V$ and for deciding projectivity:

• Algorithm 1 (cercafan): Exhaustively checks all possibilities combinatorially but is exponential in complexity.
• Algorithm 2 (G-cercafan): Utilizes the Gröbner fan of the toric ideal associated to $V$, with output matching the set of projective fans. This method is efficient for higher dimensions and Picard numbers.

A critical structural result is the correspondence between chambers of the secondary fan of $V$ and Gröbner cones of the toric ideal, refined by the Gröbner fan. Projectivity of a toric $\mathbf{Q}$-factorial variety is thus translated into a criterion about the position of the nef cone in the space of divisor classes (Rossi et al., 2020).

6. Applications and Phenomena in Families: Nef Cone Jump and Open Problems

Families of projective $\mathbf{Q}$-factorial varieties may exhibit jumps in the dimension of their nef cones as parameters vary. In rationally parametrized toric families, such as those examined by Berchtold–Hauen and further studied in the context of secondary fans, one observes that a limit variety may lose projectivity while retaining $\mathbf{Q}$-factoriality, or conversely, a variety may gain a nontrivial nef cone at a special fiber (Rossi et al., 2020).

Open problems include the pseudofan conjecture, combinatorial recovery of non-projective fans from algebraic data, and studying monomial initial ideals and Alexander duality in the non-radical case.

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Summary Table: Key Defect and Factoriality Criteria

Singularities/Class	Defect Formula	$\mathbf{Q}$-factorial $\Rightarrow$ factorial?	Projective Criterion
LCI, codim Sing $\geq$3	$\sigma(X) = h^{2n-2} - h^2$	Yes	Nef cone full dimensionality ($\dim = r$)
Cohen–Macaulay, codim Sing $\geq$4	$\sigma(X) = h^{2n-2} - h^2$ (mod torsion)	In general, no ($\mathbf{Q}$-factorial only)	Strictly convex support function exists
Toric, $\mathbf{Q}$-factorial	Maximal cones simplicial	N/A	Chamber in secondary fan full dimensional

This synthesis draws upon the results and constructions from (Jung et al., 19 Jan 2026) (Jung–Saito) and (Rossi et al., 2020), encompassing both cohomological and combinatorial treatments of projective factorial and $\mathbf{Q}$-factorial varieties.