An obstruction to smoothing stable maps

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.

💡 Research Summary

The paper introduces a new obstruction to smoothing stable maps in smooth projective varieties, extending previously known criteria. After recalling the basic definitions of prestable curves, stable maps, ghost components (the union of irreducible components on which the map is constant) and effective sub‑curves (the complement of the ghost), the authors focus on the local geometry at points where a ghost component meets the effective part.

For a smooth closed point (p) on a prestable curve (C) they consider the short exact sequence
(0\to\mathcal O_C\to\mathcal O_C(p)\to T_{C,p}\to0)
and the induced connecting homomorphism (\delta_{C,p}:T_{C,p}\to H^1(C,\mathcal O_C)). By Serre duality this map corresponds to the evaluation map (ev_{C,p}:H^0(C,\omega_C)\to T_{C,p}^\vee). They also denote by (df_p:T_{C,p}\to T_{X,f(p)}) the differential of the stable map at (p).

The main result (Theorem 1.7) states that if a non‑constant stable map (f:C\to X) is eventually smoothable—i.e. after possibly embedding (X) into a larger smooth projective variety the map becomes a limit of maps from smooth domains—then for every ghost component (C) meeting the effective sub‑curve (E) the linear map

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