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Coble surfaces: projective models and automorphisms with related topics

Federico Pieroni

Abstract

In this work, we want to show several properties of an unnodal, complex Coble surface $X$ with irreducible boundary curve $C \in |-2 K_X|$. Namely, we show that every isotropic sequence ${\cal E}_1, \ldots, {\cal E}_r$ with $r \le 8$ and ${\cal E}_i {\cal E}_j = 1 - δ_{i, j}$ can be extended to a sequence of length $10$. Moreover, such a surface admits a birational quintic model $\overline X \subset \mathbb{P}^3$, with equation $αX_0 X_1^2 X_2^2 + βX_0 X_1^2 X_3^2 + γX_0 X_2^2 X_3^2 + X_1 X_2 X_3 q = 0$, where $q$ is a quadric form. Finally, we use this birational model to show that every biregular involution $i : X \to X$ on such a Coble surface is the lift of a Bertini involution.

Coble surfaces: projective models and automorphisms with related topics

Abstract

In this work, we want to show several properties of an unnodal, complex Coble surface with irreducible boundary curve . Namely, we show that every isotropic sequence with and can be extended to a sequence of length . Moreover, such a surface admits a birational quintic model , with equation , where is a quadric form. Finally, we use this birational model to show that every biregular involution on such a Coble surface is the lift of a Bertini involution.

Paper Structure

This paper contains 31 sections, 65 theorems, 601 equations, 3 figures.

Table of Contents

  1. Generalities on surfaces
  2. Line bundles, divisors and intersection theory on surfaces
  3. Projective models via line bundles
  4. Rational surfaces
  5. Hirzebruch surfaces
  6. Blow ups of $\mathbb{P}^2$
  7. Del Pezzo surfaces
  8. Del Pezzo surfaces of degree in $\{3, \ldots, 8\}$
  9. Del Pezzo surfaces of degree $2$, and the Geiser involution
  10. Del Pezzo surfaces of degree $1$, and the Bertini involution
  11. The De-Jonquieres involution
  12. Some Lattice theory
  13. Enriques surfaces
  14. Coble surfaces
  15. First properties and examples
  16. ...and 16 more sections

Key Result

Theorem 1

If $D$ is any divisor on a smooth variety $X$ of dimension $n$, then for any $k \in \{0, \ldots, n\}$ we have

Figures (3)

  • Figure 1: A $(-1)$-chain with length $4$. The blue component $F_4$ is a $(-1)$-curve, the red components are $(-2)$-curves.
  • Figure 2: With the same colors as above, the set $E_1', E_2', E_3', E_4'$ is an extension of $E_1, E_2$. The extra components are dashed.
  • Figure 3: We can imagine to contract the $(-1)$-chain $F_1 + \cdots + F_r$ on the point $p$ starting from the $(-1)$-curve $F_r$. This turns $F_{r - 1}$ into a $(-1)$-curve, so it can be contracted as well, and we proceed backwards to $F_1$. Each step makes the self - intersection $K_X^2$ jump by $1$, hence we get formula \ref{['length']}. Meanwhile, the self - intersection $\tilde{D}^2$ is affected only by $F_s, F_{s - 1}, \ldots, F_1$, and each of these produces a jump by $1$. The sum of the contributions of all $(-1)$-chains involved gives formula \ref{['diff']}.

Theorems & Definitions (109)

  • Theorem 1: Serre duality
  • Definition 2
  • Theorem 3: Nakai-Moishezon Criterion
  • Theorem 4: Hodge Index Theorem
  • Theorem 5: Riemann-Roch Theorem on surfaces
  • Theorem 6: Reider's Theorem
  • Proposition 7
  • Definition 8
  • Theorem 9
  • Definition 10
  • ...and 99 more