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On Subvarieties of Degenerations of Fano Varieties

Santai Qu

Abstract

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fibre.

On Subvarieties of Degenerations of Fano Varieties

Abstract

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fibre.

Paper Structure

This paper contains 14 sections, 9 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Varieties and schemes
  4. Contractions
  5. Divisors and singularities of pairs
  6. Relatively dense subset and unirationality
  7. Grassmannians
  8. Rational-connectivity and separable-rational-connectivity
  9. Prime regular discrete valuation rings
  10. Ax conjecture in positive characteristics
  11. Nice DVRs
  12. Preliminary results
  13. Dimensions of subvarieties on degenerations
  14. Examples

Key Result

Theorem 1.1

Let $k$ be a field of characteristic 0, $C$ a smooth $k$-curve, $Z$ a reduced, irreducible, projective $k$-variety, and $g\colon Z\to C$ a $k$-morphism. Assume that the generic fibre $F_{\text{gen}}$ is Let $c\in C$ be a closed point with residue field $k (c)$. Then, the fibre $g^{-1}(c)$ contains a $k(c)$-subscheme which is geometrically irreducible. If, in addition, every $k(c)$-irreducible com

Theorems & Definitions (28)

  • Theorem 1.1: Kollarcom
  • Corollary 1.2: Kollarcom
  • proof
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1: hx07
  • Definition 2.2: cf. hx07
  • Definition 2.3
  • Definition 2.4: deJongStarrofsections and Kollar_rational_curves
  • ...and 18 more