A short proof of the Goettsche conjecture

Martijn Kool (Imperial College London)

Monday 15th November, 2010 16:00-17:00 204


Counting the number of curves of degree d with n nodes (and no further singularities) going through (d^2+3d)/2 - n points in general position in the projective plane is a problem which was already considered more than 150 years ago. More recently, people conjectured that for sufficiently large d this number should be given by a polynomial of degree 2n in d. More generally, the Goettsche conjecture states that the number of n-nodal curves in a general n-dimensional linear subsystem of a sufficiently ample line bundle L on a nonsingular projective surface S is given by a universal polynomial of degree n in the 4 topological numbers L^2, L.K_S, (K_S)^2 and c_2(S). In a joint work with Vivek Shende and Richard Thomas, we give a short (compared to existing) proof of this conjecture.

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