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Descend of morphisms of varieties

Supravat Sarkar

TL;DR

The paper addresses when a dominant morphism $f:X→W$ that is constant on the fibers of another dominant morphism $φ:X→Y$ descends to $Y$. It introduces a descent mechanism in positive characteristic using iterated Frobenius twists, producing a dominant rational map $h:Y→W_N$ with $F_N\circ f=h\circ φ$, where $W_N$ is either $W$ or its Frobenius twist depending on whether $K(Y)$ embeds purely inseparably into $K(X)$. Under the assumptions that $Y$ is normal and $\, ext{dim} \, \overline{(Y\setminus φ(X))}\le \text{dim }Y-2$, several equivalent formulations of descent are established, including morphism-level descent, continuous descent, and a local-affine descent criterion; these equivalences are further guaranteed when $W$ is affine or when $φ$ is a closed or surjective open map. The arguments combine generic-fiber techniques, normal closures, and invariant theory of function fields, along with a local extension lemma that ensures morphism descent from rational data. An application shows descent under closed maps and pure inseparable closure, extending classical contraction results, while an example highlights the limits of the equivalences in general settings.

Abstract

Given varieties $X, Y, W$ and dominant morphisms $φ:X\to Y$ and $f:X\to W$ such that $f$ is constant on fibres of $φ$ , we give sufficient conditions to guarantee that $f$ descends to a rational map or a morphism $Y\to W.$ We pay special attention to the case that the ground field has positive characteristic. This extends previous works of Aichinger and Das, who proved similar results for some classes of affine varieties.

Descend of morphisms of varieties

TL;DR

The paper addresses when a dominant morphism that is constant on the fibers of another dominant morphism descends to . It introduces a descent mechanism in positive characteristic using iterated Frobenius twists, producing a dominant rational map with , where is either or its Frobenius twist depending on whether embeds purely inseparably into . Under the assumptions that is normal and , several equivalent formulations of descent are established, including morphism-level descent, continuous descent, and a local-affine descent criterion; these equivalences are further guaranteed when is affine or when is a closed or surjective open map. The arguments combine generic-fiber techniques, normal closures, and invariant theory of function fields, along with a local extension lemma that ensures morphism descent from rational data. An application shows descent under closed maps and pure inseparable closure, extending classical contraction results, while an example highlights the limits of the equivalences in general settings.

Abstract

Given varieties and dominant morphisms and such that is constant on fibres of , we give sufficient conditions to guarantee that descends to a rational map or a morphism We pay special attention to the case that the ground field has positive characteristic. This extends previous works of Aichinger and Das, who proved similar results for some classes of affine varieties.

Paper Structure

This paper contains 4 sections, 2 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Proof of $(1)$
  3. Proof of $(2)$:
  4. Application and an example

Key Result

Lemma 3.1

Let be a commutative diagram of varieties over an algebraically closed field $k$, $f$ and $\phi$ are dominant morphisms, $h$ a dominant rational map. Suppose $Y$ is normal, $W$ is affine, and $\dim \overline{(Y\setminus\phi(X))}\leq \dim Y-2$. Then $h$ is a morphism.

Theorems & Definitions (4)

  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof