An obstruction to smoothing stable maps

TL;DR

We address smoothing stable maps $f:\mathscr{C}\to X$ to a smooth projective variety $X$, and introduce a local obstruction arising from ghost components. The approach analyzes a smoothing family near ghost components to extract a leading-term rational map $G_m:\tilde{\mathscr{S}}\dashrightarrow T_{X,q}$ with prescribed simple poles and residues, connecting this analytic data to a cohomological obstruction. The main result shows that if a stable map is eventually smoothable, then the kernel of the linear map $\bigoplus_{p\in C\cap\mathscr{E}} \delta_{C,p}\otimes d(f|_{\mathscr{E}})_p: \bigoplus_{p\in C\cap\mathscr{E}} T_{C,p}\otimes T_{\mathscr{E},p} \to H^1(C,\mathcal{O}_C)\otimes T_{X,q}$ is nontrivial, and a rank-based corollary gives a more accessible criterion independent of global geometry. The paper also provides concrete genus-one and higher-genus examples demonstrating the strength of the obstruction and highlighting the local nature of the phenomenon. These obstructions inform potential compactifications of the map moduli that are smaller than the usual stable-map compactification and clarify limits of smoothing techniques in higher genus. Overall, the work advances a local-to-global understanding of when stable maps can be smoothed, with explicit constructions and a robust coordinate- and deformation-theoretic framework.

Abstract

We describe an obstruction to smoothing stable maps in smooth projective varieties, which generalizes some previously known obstructions. Our obstruction comes from the non-existence of certain rational functions on the ghost components, with prescribed simple poles and residues.

Figures (6)

• Figure 1: An example of a stable map which is not eventually smoothable, obtained by taking $N=3$, $h=4$ and $n=Nh=12$ in §\ref{['subsubsec:example-1']}. For $1\le i\le 3$, the curves $\mathscr{E}_{i,1},\ldots,\mathscr{E}_{i,4}$ are attached at four general points of $C$.
• Figure 2: An example of a non-eventually smoothable stable map $f_{\sigma}$ for $h = 3$ and $\sigma=(231)\in\mathfrak{S}_3$ as in Lemma \ref{['Lem: sigmaNoFixedPoint']}.
• Figure 3: The smoothable stable map $f_{\text{id}} = g\circ\psi$ from Lemma \ref{['Lem: sigma=ID']} in the case where $h=3$.
• Figure 4: Resolving the $A_l$-singularity $xy=t^{l+1}$ by repeatedly blowing up singular points. At each of the $l+1$ nodes of the $t = 0$ fibre of the resolved surface, the labels specify a pair of local coordinates whose product is $t$.
• Figure 5: Resolving the singularities of $\mathscr{S}$ by repeatedly blowing up the singular points to get $\tilde{\mathscr{S}}$, with exceptional curves $E_{i,j}$.

Theorems & Definitions (30)

• Definition 1.1: Prestable curve, stable map
• Definition 1.2: Ghost, effective sub-curve
• Definition 1.3: Eventual smoothability
• Remark 1.4
• Definition 1.5: Residue obstruction, evaluation map
• Definition 1.6: First derivative
• Theorem 1.7: Obstruction to eventual smoothability
• Corollary 1.8
• proof
• Remark 1.9