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Flexible affine cones and flexible coverings

Matheusz Michałek, Alexander Perepechko, Hendrik Süß

Abstract

We provide a new criterion for flexibility of cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre--Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces.

Flexible affine cones and flexible coverings

Abstract

We provide a new criterion for flexibility of cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre--Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces.

Paper Structure

This paper contains 8 sections, 30 theorems, 39 equations, 1 figure.

Table of Contents

  1. Flexibility of affine cones
  2. Secant of Segre--Veronese variety
  3. The combinatorial description of T-varieties
  4. Cones over projective T-varieties
  5. Total coordinate spaces
  6. Del Pezzo surfaces
  7. Smooth complexity-one T-varieties
  8. Flexibility of total coordinate spaces vs. flexible coverings

Key Result

Theorem \oldthetheorem

Let $X$ be an irreducible affine variety of dimension $\ge 2$. Then the following conditions are equivalent.

Figures (1)

  • Figure 1: An f-divisor

Theorems & Definitions (83)

  • Theorem \oldthetheorem: AFKKZ
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 73 more