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Variation of cones of divisors in a family of varieties -- Fano type case

Sung Rak Choi, Zhan Li, Chuyu Zhou

TL;DR

This paper analyzes how cones of divisors in a family of varieties behave when fibers are Fano type on a dense set. After suitable base changes and shrinking, it proves deformation invariance for the Néron-Severi space and for the effective, nef, and movable cones, as well as the Mori chamber decomposition, and shows that the minimal model program behaves uniformly across fibers. These deformation results yield a partial answer to whether fiberwise Fano type persists globally in families and enable a boundedness statement for birational contraction models of Fano type varieties. The findings provide a framework connecting fiberwise geometry to global moduli questions and extend to corollaries for Mori dream spaces and complements in the Fano type setting.

Abstract

We investigate the relationship between the Fano type property on fibers over a Zariski dense subset and the global Fano type property. We establish the invariance of Néron-Severi spaces, nef cones, effective cones, movable cones, and Mori chamber decompositions for a family of Fano type varieties after a generically finite base change. Additionally, we show the uniform behavior of the minimal model program for this family. These results are applied to the boundedness problem of Fano type varieties.

Variation of cones of divisors in a family of varieties -- Fano type case

TL;DR

This paper analyzes how cones of divisors in a family of varieties behave when fibers are Fano type on a dense set. After suitable base changes and shrinking, it proves deformation invariance for the Néron-Severi space and for the effective, nef, and movable cones, as well as the Mori chamber decomposition, and shows that the minimal model program behaves uniformly across fibers. These deformation results yield a partial answer to whether fiberwise Fano type persists globally in families and enable a boundedness statement for birational contraction models of Fano type varieties. The findings provide a framework connecting fiberwise geometry to global moduli questions and extend to corollaries for Mori dream spaces and complements in the Fano type setting.

Abstract

We investigate the relationship between the Fano type property on fibers over a Zariski dense subset and the global Fano type property. We establish the invariance of Néron-Severi spaces, nef cones, effective cones, movable cones, and Mori chamber decompositions for a family of Fano type varieties after a generically finite base change. Additionally, we show the uniform behavior of the minimal model program for this family. These results are applied to the boundedness problem of Fano type varieties.

Paper Structure

This paper contains 12 sections, 20 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Families with fibers of Fano type varieties over Zariski dense subsets
  4. Deformation of Néron-Severi spaces and nef cones
  5. Deformation of Néron-Severi spaces
  6. Deformation of nef cones
  7. MMP in a family and deformations of effective cones and Mori chamber decompositions
  8. Stable boundedness
  9. MMP in a family
  10. Deformation of effective cones
  11. Deformation of movable cones and Mori chamber decompositions
  12. Boundedness of birational models of Fano type varieties

Key Result

Theorem 1.2

Let $f: X \to T$ be a projective, surjective morphism between varieties such that $f_*\mathcal{O}_X=\mathcal{O}_T$. Suppose that there exists a Zariski dense subset $S\subset T$ such that fibers $X_s, s\in S$ are Fano type varieties. Then, after shrinking $T$, there exists a divisor $D$ on $X$ such Furthermore, if $X_s, s\in S$ fall into one of the following cases: then, after shrinking $T$, we

Theorems & Definitions (40)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm: FT is constructible']}
  • Lemma 4.1
  • ...and 30 more