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Fano 4-folds with $b_2>12$ are products of surfaces

Cinzia Casagrande

Abstract

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We show that if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f: X->Y such that the image S of the exceptional divisor is a surface, together with the author's previous work on Fano 4-folds. In particular, given f: X->Y as above, under suitable assumptions we show that S is a smooth del Pezzo surface with -K_S given by the restriction of -K_Y.

Fano 4-folds with $b_2>12$ are products of surfaces

Abstract

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We show that if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f: X->Y such that the image S of the exceptional divisor is a surface, together with the author's previous work on Fano 4-folds. In particular, given f: X->Y as above, under suitable assumptions we show that S is a smooth del Pezzo surface with -K_S given by the restriction of -K_Y.
Paper Structure (5 sections, 22 theorems, 18 equations, 1 figure)

This paper contains 5 sections, 22 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Showing that $S$ is a del Pezzo surface
  5. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $X$ be a smooth Fano $4$-fold with $\rho_X> 12$. Then $X\cong S_1\times S_2$, where $S_i$ are del Pezzo surfaces.

Figures (1)

  • Figure 3.5: The varieties in Proposition \ref{['oneray']}.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: morimukai2, Theorem 1.2
  • Theorem 1.4: codim, Theorem 3.3
  • Theorem 1.5: delta3, Proposition 1.5
  • Theorem 1.6: small, Theorem 1.1
  • Theorem 2.1: AW, Theorem on p. 256
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 28 more