On deformations of curves supported on rigid divisors

TL;DR

This paper investigates the structure of fibred surfaces supported on rigid divisors, developing a formalism for families of varieties and constructing a global adjoint map to produce supporting divisors, thus addressing a conjecture of Xiao.

Contribution

It introduces a new formalism for families of varieties and constructs a global adjoint map to analyze supporting divisors, advancing understanding of fibred surfaces supported on rigid divisors.

Findings

Established a structure theorem for fibred surfaces supported on rigid divisors.

Developed a formalism for one-dimensional families of varieties of any dimension.

Constructed a global adjoint map to produce supporting divisors.

Abstract

Motivated by a conjecture of Xiao, we study supporting divisors of fibred surfaces. On the one hand, after developing a formalism to treat one-dimensional families of varieties of any dimension, we give a structure theorem for fibred surfaces supported on relatively rigid divisors. On the other hand, we study how to produce supporting divisors by constructing a global adjoint map for a fibration over a curve (generalizing the infinitesimal constructions of Collino, Pirola, Rizzi and Zucconi).