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$q$-bic threefolds and their surface of lines

Raymond Cheng

Abstract

For any power $q$ of the positive ground field characteristic, a smooth $q$-bic threefold -- the Fermat threefold of degree $q+1$ for example -- has a smooth surface $S$ of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of $S$. Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute cohomology of the structure sheaf of $S$ when $q$ is prime.

$q$-bic threefolds and their surface of lines

Abstract

For any power of the positive ground field characteristic, a smooth -bic threefold -- the Fermat threefold of degree for example -- has a smooth surface of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of . Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute cohomology of the structure sheaf of when is prime.
Paper Structure (35 sections, 69 theorems, 245 equations, 1 figure)

This paper contains 35 sections, 69 theorems, 245 equations, 1 figure.

Table of Contents

  1. q-bic hypersurfaces
  2. q-bic forms and hypersurfaces
  3. Classification
  4. q-bic points and curves
  5. Smoothness
  6. Hermitian structures
  7. Scheme of lines
  8. q-bic surfaces
  9. q-bic threefolds
  10. Cone situation
  11. Cone situation
  12. Examples
  13. Closure of T^o
  14. Smooth cone situation
  15. q-bic threefolds of type 1³N₂
  16. ...and 20 more sections

Key Result

Theorem 1

The scheme $S$ of lines of a smooth $q$-bic threefold $X$ is an irreducible, smooth, projective surface of general type. The Fano correspondence $S \leftarrow \mathbf{L} \rightarrow X$ induces purely inseparable isogenies amongst supersingular abelian varieties of dimension $\frac{1}{2}q(q-1)(q^2+1)$.

Figures (1)

  • Figure 1: The dimensions of the $\mathrm{H}^0(C,\mathcal{F}_i)$ are displayed with $q = 8$ on the left, and $q = 9$ on the right. The numbers are arranged so that the first row displays the dimensions of $\mathrm{H}^0(C,\mathcal{F}_i)$ for $0 \leq i \leq q-1$. These were obtained from computer calculations done with Macaulay2M2.

Theorems & Definitions (125)

  • Theorem
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 1.2
  • proof
  • Lemma 1.7
  • proof
  • Lemma 1.11
  • ...and 115 more