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A construction of smooth varieties admitting small contractions

Yuto Masamura, Tomoki Yoshida

TL;DR

This work develops a two-step blowup strategy to construct smooth varieties with small contractions, extending Kawamata’s fourfold examples. By analyzing the relative nef cone and the cone of curves, the authors establish precise conditions under which the blowup along centers $A''$ and then the strict transform of $B''$ yields a $K_X$-extremal small contraction, with the target $X_0$ identifiable as the blowup of $X''$ along $A''\cup B''$, and describe flips when $a<b$. They then derive global nefness criteria for divisors on the resulting spaces, including a general product-case yielding explicit nef cones generated by $H_1$, $H_2$, $H_1+dH_2-E$, and $H_1+dH_2-E-F$. Applying this to the product of two del Pezzo surfaces, they construct smooth weak Fano fourfolds with Picard number up to 8 that admit a small contraction, classify when these fourfolds are Fano, weak Fano, or of Fano type, and show explicit non-product examples. The results supply new explicit fourfolds with small contractions and illuminate the interplay between blowups, nef cones, and MMP contractions in higher dimensions.

Abstract

We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to be nef. As an application, we obtain weak Fano fourfolds from products of two del Pezzo surfaces.

A construction of smooth varieties admitting small contractions

TL;DR

This work develops a two-step blowup strategy to construct smooth varieties with small contractions, extending Kawamata’s fourfold examples. By analyzing the relative nef cone and the cone of curves, the authors establish precise conditions under which the blowup along centers and then the strict transform of yields a -extremal small contraction, with the target identifiable as the blowup of along , and describe flips when . They then derive global nefness criteria for divisors on the resulting spaces, including a general product-case yielding explicit nef cones generated by , , , and . Applying this to the product of two del Pezzo surfaces, they construct smooth weak Fano fourfolds with Picard number up to 8 that admit a small contraction, classify when these fourfolds are Fano, weak Fano, or of Fano type, and show explicit non-product examples. The results supply new explicit fourfolds with small contractions and illuminate the interplay between blowups, nef cones, and MMP contractions in higher dimensions.

Abstract

We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to be nef. As an application, we obtain weak Fano fourfolds from products of two del Pezzo surfaces.
Paper Structure (10 sections, 24 theorems, 102 equations)

This paper contains 10 sections, 24 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. General construction of smooth varieties admitting small contractions
  3. Nef divisors on two successive blowups
  4. Applications for smooth weak Fano fourfolds
  5. Preliminaries
  6. Conormal sheaves
  7. Nefness of divisors
  8. Nefness of divisors on blowups
  9. General construction of varieties admitting small contractions
  10. Applications for products of two del Pezzo surfaces

Key Result

Theorem A

Let $X"$ be a smooth quasi-projective variety of dimension $\ge3$, and let $A",B"\subseteq X"$ be smooth subvarieties of codimension $a,b\ge2$. Assume that Let $X$ be the blowup of $X"$ successively along $A"$ and along the strict transform of $B"$. Then there exists a birational contraction $\psi\colon X\to X_0$ over $X"$ with $\rho(X/X_0)=1$. Moreover,

Theorems & Definitions (43)

  • Theorem A: \ref{['Theorem: existence of small contraction']}
  • Theorem B: \ref{['Theorem: existence of small contraction', 'corollary: flip']}
  • Theorem C: \ref{['lemma: nefness of H-E', 'lemma: nefness of H-E-F']}
  • Theorem D: casagrande_2025_towards_the_classification_of_Fano_4-folds_with_b2_7
  • Theorem E: \ref{['cor: characterization of Fano']}
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • ...and 33 more