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Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5

Alexander Perepechko

TL;DR

The paper proves that affine cones over del Pezzo surfaces of degrees $5$ and $4$ are flexible, implying infinite transitivity of the special automorphism group on their smooth loci. It achieves this by developing a transversal-cover criterion based on open cylinders on the base surface and linking each cylinder to a homogeneous ${\mathbb G}_a$-action on the cone via Zariski-type constructions. For degree $5$, the authors construct 15 cylinders tied to five disjoint $(-1)$-curve blowdowns and show these cylinders are $H$-polar for all ample $H$, yielding a full transversal cover and flexibility for any very ample polarization. For degree $4$, they build five cylinder families from a five-cycle of $(-1)$-curves, obtain a transversal cover, and compute the corresponding $72$-ray subcone of the ample cone consisting of $H$-polar divisors; while it includes $-K_Y$, it does not exhaust the entire ample cone, leaving openness on the flexibility for arbitrary very ample divisors.

Abstract

We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive.

Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5

TL;DR

The paper proves that affine cones over del Pezzo surfaces of degrees and are flexible, implying infinite transitivity of the special automorphism group on their smooth loci. It achieves this by developing a transversal-cover criterion based on open cylinders on the base surface and linking each cylinder to a homogeneous -action on the cone via Zariski-type constructions. For degree , the authors construct 15 cylinders tied to five disjoint -curve blowdowns and show these cylinders are -polar for all ample , yielding a full transversal cover and flexibility for any very ample polarization. For degree , they build five cylinder families from a five-cycle of -curves, obtain a transversal cover, and compute the corresponding -ray subcone of the ample cone consisting of -polar divisors; while it includes , it does not exhaust the entire ample cone, leaving openness on the flexibility for arbitrary very ample divisors.

Abstract

We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive.

Paper Structure

This paper contains 8 sections, 4 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Flexibility of affine cones
  3. Del Pezzo surface of degree 5
  4. Cylinders
  5. Polarity condition
  6. Del Pezzo surfaces of degree 4
  7. Cylinders
  8. Polarity condition

Key Result

Theorem 1

Let $X$ be an affine algebraic variety of dimension $\ge2$. Then the following conditions are equivalent:

Figures (3)

  • Figure 1: Arrangement of cylinders on the incidence graph of $(-1)$-curves on the del Pezzo surface of degree 5. The gray and the black vertices correspond to $(-1)$-curves forming the complement to a cylinder. The dashed edges correspond to $(-1)$-curve intersections contained in the cylinder. The double edge corresponds to the base point of the cylinder.
  • Figure 2: Black vertices correspond to 4-tuples of $(-1)$-curves. Every blowing down defines three cylinders similarly as on fig. \ref{['graph5-3']}.
  • Figure 3: The incidence graph of $(-1)$-curves on a del Pezzo surface of degree 4. On the left the gray vertex corresponds to the quadric preimage $C_1$ and black vertices correspond to the contracted $(-1)$-curves. The dashed edges correspond to $(-1)$-curve intersections contained in the cylinders of a family. Four other families corresponding to $C_2,\ldots, C_5$ are obtained symmetrically by the graph rotations.

Theorems & Definitions (8)

  • Theorem 1: flex
  • Definition 2: Z
  • Definition 3
  • Definition 4
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7