Constructing smoothings of stable maps

TL;DR

This work studies when stable maps $f:\mathscr{C}\to X$ can be obtained as limits of maps from smooth curves after embedding $X$ into some $\mathbb{P}^N$, introducing the notion of eventual smoothability. It develops a local-to-global deformation-theoretic framework: a non-constant stable map is eventually smoothable if it is locally formally smoothable, and it provides a large, explicit class of maps—stable maps with model ghosts—that satisfy this local criterion. The authors construct explicit model pinching families from data on a curve $C$ with marked points $p_i$ and coprime integers $d_i$, yielding model singularities $\mathscr{Y}_0$ and corresponding smoothings. They then prove that stable maps factoring through such model pinchings are eventually smoothable, and illustrate the theory with concrete genus and marked-point examples, including connections to Weierstrass semigroups, canonical divisors, and suspension constructions. The results generalize to a wide class of maps and provide concrete tools for constructing sharp compactifications and understanding obstructions in broader settings, with implications for Gromov–Witten theory and related enumerative problems. Further work will address obstructions to smoothability and refine the sharp-compactification framework in higher genus.

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.

Figures (18)

• Figure 1: The model family of stable maps, from §\ref{['subsec:proof-strategy']}, associated to $(C,p)$.
• Figure 2: Pinching morphism (Definition \ref{['def:pinching']}). The first-order tangency of the two branches of $\mathscr{S}$ means that $C$ is of genus $1$.
• Figure 3: Depiction of $\mathscr{C}$ (left) and $\mathscr{S}$ (right), from Example \ref{['exa:genus-0-pinching']}, in the case $n=3$. The curve $\Sigma_i$ and its image in $\mathscr{S}$ are in purple, while the curve $C_i$ and its image $s_i$ are in green. Note that $\mathscr{C}$ can be embedded into a non-singular surface while $\mathscr{S}$ cannot (its embedding dimension at $s_i$ is $3$).
• Figure 4: Depiction of the objects appearing in Remark \ref{['rem:etale-base-change-pinching']}. In the rightmost column, corresponding to $\nu_{U_i,\varphi_i}$, $C_i$ (top) and $s_i'$ (bottom) are shown in green while $\tilde{V}_i$ (top) and $V_i$ (bottom) are shown in purple.
• Figure 5: Depiction of the set up for the proof of Theorem \ref{['thm:local-criterion-for-smoothability']}. The ghost sub-curve $C$ is shown in green, while $\mathscr{C}\setminus C = \mathscr{S}\setminus\{s_1,\ldots,s_r\}$ is shown in purple. Internal (resp. external) nodes are shown in yellow (resp. blue).

Theorems & Definitions (144)

• Definition 1.1: Prestable curve, stable map
• Definition 1.2: Eventual smoothability
• Theorem A: see Theorem \ref{['thm:local-criterion-for-smoothability']}, Local criterion for eventual smoothability
• Theorem B: see Theorem \ref{['thm:model-ghost-implies-eventual-smoothability']}
• Example 1.3: see §\ref{['subsubsec:all-d-bigger-than-1']}, specifically Example \ref{['exa:genus-3-with-2-points-d=2,3']}, for more details
• Example 1.4: Follows from Zinger-sharp-compactness
• Example 1.5: Follows from DW-counting or ekholm-shende-ghost
• Definition 2.2: Cotangent complex
• Definition 2.3: $T^i$ modules
• Remark 2.4