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Nets of quadric surfaces and plane cubics and their GIT stability

Masafumi Hattori, Theodoros Stylianos Papazachariou, Aline Zanardini

Abstract

A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via the Gale transformation of the remaining seven base points. To both nets, one can also naturally associate the same smooth plane quartic. In this paper, we generalize the cycle of correspondences arising from nets of quadrics that define rational elliptic threefolds and provide a complete criterion for GIT stability of the three underlying geometric objects using birational-geometric techniques.

Nets of quadric surfaces and plane cubics and their GIT stability

Abstract

A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via the Gale transformation of the remaining seven base points. To both nets, one can also naturally associate the same smooth plane quartic. In this paper, we generalize the cycle of correspondences arising from nets of quadrics that define rational elliptic threefolds and provide a complete criterion for GIT stability of the three underlying geometric objects using birational-geometric techniques.
Paper Structure (15 sections, 30 theorems, 36 equations, 1 table)

This paper contains 15 sections, 30 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and notations
  3. The relevant geometric invariant theory
  4. GIT for linear systems of hypersurfaces
  5. The log canonical threshold and toric valuations
  6. A cycle of correspondences and Gale duality
  7. From nets of quadrics to nets of cubics and back
  8. Generalized Gale transformation and good nets
  9. The associated rational elliptic threefolds
  10. GIT stability of linear systems of plane cubics
  11. The proof of our main theorem
  12. Further questions
  13. GIT unstability of nets of quadrics in P3
  14. An overview of the algorithm
  15. Stability criteria and the discriminant quartic

Key Result

Theorem 1.1

Let $\mathcal{N}$ be a net of quadrics in $\mathbb{P}^3$ with a discriminant quartic $\Delta(\mathcal{N})$ that is reduced and has only ADE singularities. Then, any choice of a base point $p \in \mathbb{P}^3$ of $\mathcal{N}$ defines a net of plane cubics $G(\mathcal{N},p)$ with seven base points. M

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.1.1: = Corollary \ref{['cor--final']}
  • Proposition 1.2
  • Theorem 1.3: =Theorem \ref{['thm--cubics']}
  • Corollary 1.3.1: = Corollary \ref{['cor--cubic--linear--sys']}
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: hz
  • Theorem 2.3
  • proof
  • ...and 61 more