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The number of smooth varieties in an MMP on a 3-fold of Fano type

Donghyeon Kim

TL;DR

The paper addresses the number of smooth varieties that can appear during a $D$-MMP on a threefold of Fano type, establishing the sharp bound $N \le 1 + h^1(X,2D)$. It develops a four-step birational approach using flips, Leray spectral sequences, and vanishing results to relate the appearance of smooth models to a cohomological invariant, and derives a partial converse to Kodaira vanishing for movable divisors. The findings illuminate the birational structure of Fano type threefolds and the behavior of the MMP under movable divisors, with implications for nefness and semiample criteria. Overall, the work provides concrete, cohomology-driven control over the smooth outcomes of the MMP in this geometric setting.

Abstract

In this paper, we prove that for a threefold of Fano type $X$ and a movable $\mathbb{Q}$-Cartier Weil divisor $D$ on $X$, the number of smooth varieties that arise during the running of a $D$-MMP is bounded by $1 + h^1(X, 2D)$. Additionally, we prove a partial converse to the Kodaira vanishing theorem for a movable divisor on a threefold of Fano type.

The number of smooth varieties in an MMP on a 3-fold of Fano type

TL;DR

The paper addresses the number of smooth varieties that can appear during a -MMP on a threefold of Fano type, establishing the sharp bound . It develops a four-step birational approach using flips, Leray spectral sequences, and vanishing results to relate the appearance of smooth models to a cohomological invariant, and derives a partial converse to Kodaira vanishing for movable divisors. The findings illuminate the birational structure of Fano type threefolds and the behavior of the MMP under movable divisors, with implications for nefness and semiample criteria. Overall, the work provides concrete, cohomology-driven control over the smooth outcomes of the MMP in this geometric setting.

Abstract

In this paper, we prove that for a threefold of Fano type and a movable -Cartier Weil divisor on , the number of smooth varieties that arise during the running of a -MMP is bounded by . Additionally, we prove a partial converse to the Kodaira vanishing theorem for a movable divisor on a threefold of Fano type.

Paper Structure

This paper contains 3 sections, 8 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of main theorem

Key Result

Theorem 1.1

Let $X$ be a threefold of Fano type, and suppose that $D$ is a movable $\mathbb{Q}$-Cartier Weil divisor on $X$. Let be a $D$ -MMP, $D_i$ the strict transform of $D$ to $X_i$, $m$ a positive integer, and let us denote by $N$ the number of $0\le i\le n$ such that $mD_i$ is Cartier. Then we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.7: cf. Fuj17
  • ...and 7 more