… surface S and a homology class β ∈ H2(S, Z). There are various ways of counting holomorphic
curves in S in class β; … Since these are deformation invariant they must vanish in class β if …

… We calculate the stable pair theory of a projective surface S. For fixed curve class β ∈ H2(S) …
We also give conditions under which this calculates the full 3-fold reduced residue theory of …

… As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class
and … This tells us how to reduce the conditions of a double or triple point on E to tangency …

… count of translation classes of genus g curves in X of class β in X. In Section 7, we show
that Ng,β determines the full reduced descendent Gromov-Witten theory of X in genus g and …

… For more general irreducible curve classes on a K3, Yau and … in the (reduced) Gromov–Witten
or stable pairs theory of a K3 … define A1 to be the germ at the origin of the curve cut out by y …

… When we project a curve O of order N in [n] on to any lower … curve of genus 2 is the simplest
example of a class of curves … there are lc points of the curve, and we count the prime joining [r …

… Results from topological field theories such as intersections … of lower dimension in the
corresponding" spaces" of curves). … group of the surface H2 = H , X (C) and the divisor class group (…

… (not necessarily distinct) of E we define the generalized Severi variety Vd ,g(α, β, Γ) to be the
closure (in |D|) of the locus of reduced curves C in X in divisor class D of geometric genus g, …

… Another curve counting theory via stable pairs is introduced … understanding of the reduced
DT-theory. By definition, a stable … formula of counting invariants involving virtual classes. At the …

… , D(B) for a certain class of Kummer surfaces admitting a … surfaces containing a smooth
irreducible curve C with C2=2n, and no smooth irreducible curves of positive self-intersection less …