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A transfer principle for unirationality

Daniel Erman, Eric Riedl

Abstract

We apply ideas related to the strength of polynomials to provide new cases of unirational hypersurfaces. It is famously known that hypersurfaces that are smooth in very high codimension are unirational, and a simple corollary then implies that any polynomial of sufficiently high strength will give rise to a unirational hypersurface. Our main result shows that unirationality is preserved under a substitution of high collective strength. In particular, we prove that polynomials of sufficiently high secondary strength are unirational. Along the way, we introduce a ``transfer principle,'' showing that polynomials of high collective strength have Fano schemes defined by polynomials of high collective strength. This gives an alternate proof of a result of Xi Chen on unirationality of Fano schemes, and proves a weakened form of the de Jong-Debarre Conjecture. Combined with some ideas of Starr, this implies a version of Kazhdan and Ziegler's result about the universality of complete intersections of polynomials.

A transfer principle for unirationality

Abstract

We apply ideas related to the strength of polynomials to provide new cases of unirational hypersurfaces. It is famously known that hypersurfaces that are smooth in very high codimension are unirational, and a simple corollary then implies that any polynomial of sufficiently high strength will give rise to a unirational hypersurface. Our main result shows that unirationality is preserved under a substitution of high collective strength. In particular, we prove that polynomials of sufficiently high secondary strength are unirational. Along the way, we introduce a ``transfer principle,'' showing that polynomials of high collective strength have Fano schemes defined by polynomials of high collective strength. This gives an alternate proof of a result of Xi Chen on unirationality of Fano schemes, and proves a weakened form of the de Jong-Debarre Conjecture. Combined with some ideas of Starr, this implies a version of Kazhdan and Ziegler's result about the universality of complete intersections of polynomials.

Paper Structure

This paper contains 6 sections, 18 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Expected dimension of Fano scheme of $k$-planes
  4. Universality and a proof Theorem \ref{['thm:starrThm']}
  5. Unirationality of smooth complete intersections
  6. Composition of functions of high collective strength

Key Result

Theorem 1.1

Let $g_0, \dots, g_n\in \mathbb C[y_0, \dots, y_N]$ be homogeneous polynomials of degree $d$ such that $V(g_0, \dots, g_n)\subseteq \mathbb P^N$ is a complete intersection that is smooth in sufficiently high codimension (relative to $d,n$). If $F(x_0, \dots, x_n)$ is a homogeneous polynomial definin

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Fano Transfer Principle
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 40 more