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Transverse linear subspaces to hypersurfaces over finite fields

Shamil Asgarli, Lian Duan, Kuan-Wen Lai

Abstract

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.

Transverse linear subspaces to hypersurfaces over finite fields

Abstract

Ballico proved that a smooth projective variety of degree over a finite field of elements admits a smooth hyperplane section if . In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.

Paper Structure

This paper contains 12 sections, 27 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Existence of very transverse linear subspaces
  3. Strategy for proving Theorem \ref{['mainthm:tranSp']}
  4. Proof of Theorem \ref{['mainthm:tranSp']}
  5. Some numerical lemmas
  6. Smooth sections on Frobenius classical hypersurfaces
  7. Tangent hyperplanes over the ground field
  8. Interplay with Frobenius classicality
  9. Refinement of Ballico's result
  10. Bertini theorems for reduced hypersurfaces
  11. Existence of reduced hyperplane sections
  12. Transverse lines to reduced plane curves

Key Result

Theorem 1.1

Let $X\subset \mathbb{P}^n$ be a smooth hypersurface of degree $d$ defined over $\mathbb{F}_q$ and pick any $0\leq r\leq n-1$. Suppose that Then there exists a sequence of linear subspaces $H_0\subset H_1\subset \dots \subset H_r$ where each $H_i$ is an $i$-plane over $\mathbb{F}_q$ that is very transverse to $X$ in the following sense:

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 50 more