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On the p-rank of curves

Sadık Terzi

Abstract

In this paper, we are concerned with the computations of the $p$-rank of curves in two different setups. We first work with complete intersection varieties in $\mb{P}^n \text{ for}~n\ge 2$ and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. Next, we consider curves on more general surfaces with $p_g(S) = 0 = q(S)$ such as Hirzebruch surfaces and determine $p$-rank of curves on Hirzebruch surfaces.

On the p-rank of curves

Abstract

In this paper, we are concerned with the computations of the -rank of curves in two different setups. We first work with complete intersection varieties in and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. Next, we consider curves on more general surfaces with such as Hirzebruch surfaces and determine -rank of curves on Hirzebruch surfaces.

Paper Structure

This paper contains 3 sections, 3 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Complete intersections
  3. Curves in Hirzebruch surfaces

Key Result

Theorem 1

For each $i=1,\cdots,r$, we have the following isomorphism of the vector spaces where $f_j$ is a homogenous polynomial of degree $n_j$ for $j= 1,2, \cdots , i$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Proposition 6
  • proof
  • proof
  • Example 7
  • Example 8
  • ...and 1 more