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Group actions and irrationality in surface families

Nathan Chen, Louis Esser

Abstract

Rationality specializes in families of surfaces, even with mild singularities. In this paper, we study the analogous question for the degree of irrationality. We prove a specialization result when the degree of irrationality on the generic fiber arises from the quotient by a group action.

Group actions and irrationality in surface families

Abstract

Rationality specializes in families of surfaces, even with mild singularities. In this paper, we study the analogous question for the degree of irrationality. We prove a specialization result when the degree of irrationality on the generic fiber arises from the quotient by a group action.

Paper Structure

This paper contains 3 sections, 6 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Group quotients and invariant sections
  3. Proofs of main results

Key Result

Theorem 1.2

Let $\pi: {\mathcal{X}} \rightarrow B$ be a flat proper morphism over ${\mathbb C}$ to a smooth connected curve $B$. Assume that all of the fibers are integral klt surfaces. If the very general fiber ${\mathcal{X}}_b$ of $\pi$ admits a dominant rational map ${\mathcal{X}}_b \dashrightarrow \mathbb{P

Theorems & Definitions (15)

  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 5 more