Classifying additive smooth Fano toric varieties

Technical Summary: Classifying Additive Smooth Fano Toric Varieties

Problem Statement
The paper addresses the classification of additive and uniquely additive smooth Fano toric varieties over an algebraically closed field $K$ of characteristic zero. An irreducible algebraic variety $X$ of dimension $n$ is termed additive if it admits a regular action of the additive group $(K^n, +)$ with an open orbit. It is uniquely additive if this action is unique up to isomorphism. While Huang and Montero previously determined these varieties for dimension $n \le 3$, this work extends the classification to dimensions up to $n=6$. The authors aim to computationally determine existence and uniqueness for all smooth Fano toric varieties in these dimensions and provide a detailed structural classification for $n \le 4$.

Methodology
The authors rely on the correspondence between additive actions on a toric variety $X_\Sigma$ and the combinatorial properties of its associated fan $\Sigma$, specifically the Demazure roots.

1. Theoretical Framework:

   • The existence of an additive action is equivalent to the existence of a complete collection of Demazure roots (Theorem 2.4).
   • Uniqueness is determined by the condition that for a specific basis of the lattice $N$, the set of Demazure roots associated with each basis ray contains exactly one element (Theorem 2.9).
   • The authors utilize Corollaries 2.11 and 2.12, which reduce the search space by linking additivity to the existence of a maximal cone $\sigma \in \Sigma$ whose extremal rays form a basis $B$ of the lattice $N$, such that all other rays in $\Sigma$ lie in the negative octant of $B$.
   • For varieties with Picard number $\rho(X)=2$, the authors prove a general result (Proposition 3.2) that all such smooth complete toric varieties are additive, with uniqueness holding only for $\mathbb{P}^1 \times \mathbb{P}^1$.

2. Computational Implementation:

   • The authors introduce the AdditiveToricVarieties package for Macaulay2.
   • The package includes methods to compute Demazure roots (demazureRoots), check for additivity and uniqueness (additiveActions), and list classified varieties (listAdditiveSmoothFanoToricVarieties, listUniquelyAdditiveSmoothFanoToricVarieties).
   • The package leverages a pre-processed database of smooth Fano toric varieties (up to dimension 6) derived from the Graded Ring Database (GRDB) and Øbro's algorithmic classifications.
   • The computational approach involves checking the negative octant condition for maximal cones and verifying the cardinality of Demazure root sets for each ray.

Key Contributions and Results

• New Classification Data:

  • Dimension 2: All 5 isomorphism classes of smooth Fano toric surfaces are analyzed. 1 is non-additive, 2 are additive but not uniquely additive, and 2 are uniquely additive.
  • Dimension 3: Of the 18 classes, 4 are non-additive, 12 are additive (non-unique), and 2 are uniquely additive. This confirms and extends previous geometric results by Huang and Montero.
  • Dimension 4: Of the 124 classes, 45 are non-additive, 75 are additive (non-unique), and 4 are uniquely additive.
  • Dimensions 5 and 6: The authors provide counts for the first time.
    • 5-folds: 866 total classes; 396 non-additive, 466 additive (non-unique), 4 uniquely additive.
    • 6-folds: 7,622 total classes; 4,194 non-additive, 3,420 additive (non-unique), 8 uniquely additive.

• Structural Characterization:

  • The authors prove that every smooth complete toric variety with Picard number $\rho(X)=2$ is additive (Proposition 3.2).
  • They characterize uniquely additive varieties in dimensions $\le 6$ (Theorem 5.11). A smooth Fano toric variety $X$ of dimension $n \le 6$ is uniquely additive if and only if it is isomorphic to:
    1. A product of projective lines $(\mathbb{P}^1)^n$.
    2. A blow-up of $(\mathbb{P}^1)^n$ (for even $n$) at one torus-invariant point.
    3. In dimension 6, a specific variety corresponding to the polytope with ID 5816 in the GRDB.
  • This implies that uniquely additive varieties are constructed from products of $\mathbb{P}^1$ and specific blow-ups, excluding del Pezzo polytopes (which are never additive) and pseudo-del Pezzo polytopes (which are uniquely additive).

• Software Tool:

  • The AdditiveToricVarieties package is released as a functional tool for the algebraic geometry community, allowing for the automated verification of additivity and the generation of classification lists for smooth Fano toric varieties up to dimension 6.

Significance and Claims
The paper claims to provide the first complete computational classification of additive and uniquely additive smooth Fano toric varieties for dimensions up to 6. The authors note that while existential results for $n \le 3$ were previously obtained using geometric techniques, the higher-dimensional results ($n \ge 4$) and the uniqueness results for $n=3$ are new.

The work highlights the feasibility of using computational algebraic geometry to solve classification problems in higher dimensions, particularly for smooth Fano varieties, where the number of isomorphism classes grows significantly slower than that of general Gorenstein Fano varieties. The authors modestly suggest that the enumeration of uniquely additive varieties is "very simple" up to $n=6$ and propose that characterizing these classes for arbitrary $n$ and enumerating general additive complete toric varieties are natural directions for future research. They also note a connection to Ewald's conjecture regarding the embedding of smooth Fano polytopes, though they observe that not all additive varieties satisfy the conjecture's conditions computationally.