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Generic flexibility of affine cones over del Pezzo surfaces in Sagemath

Alexander Perepechko

TL;DR

This work tackles the problem of when affine cones over (weak) del Pezzo surfaces are generically flexible, parameterized by the chosen polarization. It develops a bubble-cycle–based combinatorial framework and implements a Sage module (Pe23-sage) to test the generic flexibility via the bubble Picard group and cylinder collections, leveraging open subdivisions of the Mori cone. The authors establish new degree-1 results, showing generic flexibility for certain very ample polarizations, and prove that weak del Pezzo surfaces of degree $6$ have generically flexible affine cones for all very ample polarizations, unifying several degree cases through algorithmic cylinder constructions. The approach yields practical criteria (polarity/forbidden cones, transversal cylinder collections) to certify infinite transitivity of the special automorphism group on open subsets, enhancing understanding of ${\\mathbb G}_a$-actions on affine cones and providing usable computational tools for researchers.

Abstract

Generic flexibility of affine cones over Fano varieties is a subject of active study recently. For del Pezzo surfaces the question is completely studied in degree at least 3, and partially in degree 2. We present a Sagemath module that facilitates most operations for verifying the generic flexibility of affine cones over del Pezzo surfaces and weak del Pezzo surfaces of arbitrary degree, depending on a polarization. The combinatorial approach used in this module is based on the formalism of bubble cycles and the colimit of Picard groups of blowups of the projective plane. As an example, we verify generic flexibility of affine cones over some polarizations of surfaces of degree 1 under certain conditions and over arbitrary very ample polarizations of weak del Pezzo surfaces of degree 6.

Generic flexibility of affine cones over del Pezzo surfaces in Sagemath

TL;DR

This work tackles the problem of when affine cones over (weak) del Pezzo surfaces are generically flexible, parameterized by the chosen polarization. It develops a bubble-cycle–based combinatorial framework and implements a Sage module (Pe23-sage) to test the generic flexibility via the bubble Picard group and cylinder collections, leveraging open subdivisions of the Mori cone. The authors establish new degree-1 results, showing generic flexibility for certain very ample polarizations, and prove that weak del Pezzo surfaces of degree have generically flexible affine cones for all very ample polarizations, unifying several degree cases through algorithmic cylinder constructions. The approach yields practical criteria (polarity/forbidden cones, transversal cylinder collections) to certify infinite transitivity of the special automorphism group on open subsets, enhancing understanding of -actions on affine cones and providing usable computational tools for researchers.

Abstract

Generic flexibility of affine cones over Fano varieties is a subject of active study recently. For del Pezzo surfaces the question is completely studied in degree at least 3, and partially in degree 2. We present a Sagemath module that facilitates most operations for verifying the generic flexibility of affine cones over del Pezzo surfaces and weak del Pezzo surfaces of arbitrary degree, depending on a polarization. The combinatorial approach used in this module is based on the formalism of bubble cycles and the colimit of Picard groups of blowups of the projective plane. As an example, we verify generic flexibility of affine cones over some polarizations of surfaces of degree 1 under certain conditions and over arbitrary very ample polarizations of weak del Pezzo surfaces of degree 6.
Paper Structure (10 sections, 11 theorems, 15 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Flexibility criterion
  3. Bubble cycles
  4. Bubble linear systems on ${\mathbb P}^2$
  5. Picard group and polyhedral cones
  6. Cylinder constructions
  7. Generic flexibility in degree $\ge2$
  8. Generic flexibility in degree 1
  9. Weak del Pezzo surfaces
  10. Sagemath module usage workflow

Key Result

Theorem 1.1

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

Figures (1)

  • Figure 1: The cone $B_2$ in degree 2 covered by polarity cones. A point $(a_1,a_2)$ corresponds to the divisor class $H=-K+a_1E_1+a_2E_2$. The depicted square with vertices $(0,0),(1,0),(0,1),(1,1)$ is the section of the ample part of $B_2$ by the hyperplane $-K+\bigoplus {\mathbb Q} E_i$. The filled part is covered by polarity cones.

Theorems & Definitions (42)

  • Theorem 1.1: AFKKZ
  • Definition 2.1: KPZ11
  • Definition 2.2: Pe20
  • Theorem 2.3: Pe20
  • Definition 2.4: KPZ11
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.6
  • proof
  • ...and 32 more