Constructing smoothings of stable maps

TL;DR

The paper introduces a deformation-theoretic method to identify a broad class of stable maps that can be smoothed, advancing understanding of their moduli and smoothing properties.

Contribution

It constructs a large class of stable maps with model ghosts and proves they are eventually smoothable using explicit deformation techniques.

Findings

Established a criterion for eventual smoothability of stable maps.

Constructed stable maps with model ghosts explicitly.

Proved these maps are smoothable via deformation theory.

Abstract

Let $X$ be a smooth projective variety. Define a stable map $f:C\to X$ to be "eventually smoothable" if there is an embedding $X\hookrightarrow\mathbb{P}^N$ such that $(C,f)$ occurs as the limit of a $1$-parameter family of stable maps to $\mathbb{P}^N$ with smooth domain curves. Via an explicit deformation-theoretic construction, we produce a large class of stable maps (called "stable maps with model ghosts"), and show that they are eventually smoothable.