Reduced classes and curve counting on surfaces I: theory

Details Reduced classes and curve counting on surfaces I: theory Reduced classes and curve counting on surfaces I: theory

2011-12-13

arxiv.org/abs/1112.3069v1

Mathematics

Algebraic Geometry

52 pages

Scientific paper

We give a quite general construction of reduced perfect obstruction theories for curves in a projective surface $S$. As a result we define reduced residue Gromov-Witten and stable pair invariants. The former involve twisted $\lambda$-classes and the usual insertions; the latter are combinations of virtual Euler characteristics of moduli spaces of points lying on embedded curves satisfying incidence conditions. In special cases the invariants can be considered as (a) family Gromov-Witten invariants of $S$, (b) reduced 3-fold invariants of the canonical bundle $K_S$, or (c) G\"ottsche's nodal curve counting invariants generalised to the non-ample case. In their guise (b) the two sets of invariants are related by the MNOP conjecture; we are able to verify this directly in case (c). In an Appendix written with D. Panov we show that the reduced obstruction theory for stable pairs on $S$ can be identified with one that arises from the following description of the moduli space. We first take the zero locus of a natural section of a bundle over a smooth ambient space, then we take the zero locus of a section of another bundle over that.

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