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Remarks on polynomial count varieties

Nicholas M. Katz, Fernando Rodriguez Villegas

Abstract

In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties. The two questions we address are the following. 1) If $X$ is smooth, polynomial count with $\#X(q)=q^n$ for some $n$, is $X$ isomorphic to $n$-dimensional affine space? 2) If $X$ is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~$h^{p,q}=0$ unless $p=q$? We show that in both cases the answer is no.

Remarks on polynomial count varieties

Abstract

In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties. The two questions we address are the following. 1) If is smooth, polynomial count with for some , is isomorphic to -dimensional affine space? 2) If is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~ unless ? We show that in both cases the answer is no.
Paper Structure (3 sections, 1 theorem, 34 equations)

This paper contains 3 sections, 1 theorem, 34 equations.

Table of Contents

  1. 1.
  2. 2.
  3. 3.

Key Result

Theorem 2.1

Let $X\subseteq \mathbb A^N$ be the zero locus over ${\mathbb C}$ of a polynomial $H\in {\mathbb Z}[x_1,\ldots,x_N]$ with $N\geq 1$ whose Netwon polytope $\Delta$ is equivalent to $\sigma_N$. Let where is the torus. Then the varieties $X'$ and $X$ are polynomial count outside $D$, for More precisely, and where and the $f_{r,n}$ are defined in f-defn.

Theorems & Definitions (7)

  • Theorem 2.1
  • proof
  • Example 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • proof