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Goppa Duality for Surfaces

Hikari Iwasaki

TL;DR

This work extends Gale duality from point configurations to a surface setting by developing Goppa duality for surfaces: given a finite Gorenstein scheme $\\Gamma$ of length $\gamma=r+s+2$ realized as a complete intersection on a Gorenstein surface $S$, one obtains complementary linear series $(W,W^ot)$ on $S$ whose induced maps factor the Gale pair through $S$. The special case $S=\mathbb{P}^2$ yields explicit duality statements for $(d_1,d_2)$-complete intersections and concrete geometric consequences, including Veronese surface factorization and a new proof of Coble’s theorem on four Veronese surfaces through nine general points in $\mathbb{P}^5$. The construction leverages Eisenbud–Popescu’s exact sequence framework and Koszul resolutions to produce the Goppa dual linear series; it is then applied to almost complete intersections via blowups to derive existence and, in some cases, uniqueness results for prescribed configurations, as well as connections to prior work on higher-point counts such as Deopurkar–Patel. The results provide a unifying geometric lens linking finite schemes, dual linear series, and surface embeddings, with potential generalizations to higher-dimensional bases and projective bundles, expanding the toolkit for studying interpolation by surfaces and Veronese-type embeddings.

Abstract

Gale duality is an involution of point configurations in projective spaces. Goppa duality extends this concept to a duality between linear series on a Gorenstein curve passing through prescribed points. We generalize this classical result to surfaces, establishing a duality for linear series on surfaces realizing prescribed points as a complete intersection of two divisors. We present several applications, including existence and uniqueness results for Veronese surfaces satisfying conditions to pass through given points or curves. As a key example, we give an alternative proof of Coble's result on the existence of four Veronese surfaces passing through nine general points in projective 5-space.

Goppa Duality for Surfaces

TL;DR

This work extends Gale duality from point configurations to a surface setting by developing Goppa duality for surfaces: given a finite Gorenstein scheme of length realized as a complete intersection on a Gorenstein surface , one obtains complementary linear series on whose induced maps factor the Gale pair through . The special case yields explicit duality statements for -complete intersections and concrete geometric consequences, including Veronese surface factorization and a new proof of Coble’s theorem on four Veronese surfaces through nine general points in . The construction leverages Eisenbud–Popescu’s exact sequence framework and Koszul resolutions to produce the Goppa dual linear series; it is then applied to almost complete intersections via blowups to derive existence and, in some cases, uniqueness results for prescribed configurations, as well as connections to prior work on higher-point counts such as Deopurkar–Patel. The results provide a unifying geometric lens linking finite schemes, dual linear series, and surface embeddings, with potential generalizations to higher-dimensional bases and projective bundles, expanding the toolkit for studying interpolation by surfaces and Veronese-type embeddings.

Abstract

Gale duality is an involution of point configurations in projective spaces. Goppa duality extends this concept to a duality between linear series on a Gorenstein curve passing through prescribed points. We generalize this classical result to surfaces, establishing a duality for linear series on surfaces realizing prescribed points as a complete intersection of two divisors. We present several applications, including existence and uniqueness results for Veronese surfaces satisfying conditions to pass through given points or curves. As a key example, we give an alternative proof of Coble's result on the existence of four Veronese surfaces passing through nine general points in projective 5-space.

Paper Structure

This paper contains 19 sections, 24 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Gale Transform and Goppa Duality For Curves
  3. Gale Duality
  4. Gale duality in algebro-geometric terms
  5. Classical Goppa Duality (Goppa Duality for Curves)
  6. Goppa Duality For Surfaces
  7. Goppa Duality for Higher Dimensional Locally Gorenstein Schemes
  8. When $\Gamma$ is a Complete Intersection on $S$
  9. Case $S = \mathbb{ P}^2$
  10. First example: $(3,3)$-complete intersection on $\mathbb{ P}^2$
  11. More general example: $(2,d)$- and $(3,d)$-complete intersections on $\mathbb{ P}^2$
  12. When $\Gamma$ is "Almost" a Complete Intersection
  13. Case $|R| = 1$
  14. Case $|R| = 2$
  15. Case $|R| = 3$
  16. ...and 4 more sections

Key Result

Proposition 2.1

The Gale transform of $\Gamma \subset \mathbb{ P}^r$ is well-defined and unique up to projective linear transformation on $\mathbb{ P}^r$ and $\mathbb{ P}^s$.

Theorems & Definitions (46)

  • Definition
  • Proposition 2.1
  • proof
  • Example 1
  • Lemma 2.2
  • Theorem 2.3: Gale duality, algebro-geometric version
  • Theorem 2.4: classical Goppa duality
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 36 more