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Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$

Marco Ghirlanda

TL;DR

The paper resolves the classification problem for non-toric, $\mathbb{Q}$-factorial, Gorenstein Fano threefolds of Picard number one that admit an effective $\mathbb{K}^*$-action with maximal orbit quotient $\mathbb{P}_2$. It introduces an explicit $\mathbb{T}$-variety construction via fake weighted projective spaces and a combinatorial data set $(P,g)$, and establishes precise Fano and Gorenstein criteria in terms of maximal cones and weighted data. A comprehensive, finite classification is achieved, yielding $154$ families organized into Types A,B,C, with complete Cox ring presentations and invariants such as $-\mathcal{K}_X^3$ and $HP_X(t)$ for each family. The results provide explicit models and a concrete map of the landscape of such Fano threefolds, enabling direct construction of examples and facilitating further study of their geometry via the Cox ring framework. All arguments are built around maximal orbit quotients and explicit $\mathbb{T}$-variety data to achieve a complete and non-redundant classification.

Abstract

We classify the non-toric, $\mathbb{Q}$-factorial, Gorenstein, Fano threefolds of Picard number one with an effective $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$.

Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$

TL;DR

The paper resolves the classification problem for non-toric, -factorial, Gorenstein Fano threefolds of Picard number one that admit an effective -action with maximal orbit quotient . It introduces an explicit -variety construction via fake weighted projective spaces and a combinatorial data set , and establishes precise Fano and Gorenstein criteria in terms of maximal cones and weighted data. A comprehensive, finite classification is achieved, yielding families organized into Types A,B,C, with complete Cox ring presentations and invariants such as and for each family. The results provide explicit models and a concrete map of the landscape of such Fano threefolds, enabling direct construction of examples and facilitating further study of their geometry via the Cox ring framework. All arguments are built around maximal orbit quotients and explicit -variety data to achieve a complete and non-redundant classification.

Abstract

We classify the non-toric, -factorial, Gorenstein, Fano threefolds of Picard number one with an effective -action and maximal orbit quotient .

Paper Structure

This paper contains 5 sections, 23 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Projective threefolds with a $\mathbb{K}^*$-action
  3. Gorenstein Fano threefolds with a $\mathbb{K}^*$-action
  4. Proof of Theorem \ref{['big theorem']}
  5. Classification lists

Key Result

Theorem 1.1

We obtain $154$ families of non-toric, $\mathbb{Q}$-factorial, Gorenstein Fano threefolds of Picard number one with an effective $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$. Listed according to the possible $n$ and $d_3,\dots,d_n$, the numbers of families of Type A, B, C are give Moreover, every non-toric, $\mathbb{Q}$-factorial, Gorenstein Fano threefold of Picard number one w

Theorems & Definitions (64)

  • Theorem 1.1
  • Example 1.2: A Fano threefold $X$ in a fake weighted projective space $Z$
  • Example 1.3
  • Example 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Example 2.7
  • Example 2.9
  • Example 2.11
  • ...and 54 more