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Mathematics Colloquium


Date:  Tuesday, April 16, 2013

Title:  Algebraic geometry Spring lectures: The Gottsche conjecture

Abstract:  I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques that one would never have thought of without ideas coming from string theory (Gromov-Witten invariants, BPS states) and modern geometry (the Maulik-Nekrasov-Okounkov-Pandharipande conjecture). In generic families of curves C on a complex surface S, nodal curves - those with the simplest possible singularities - appear in codimension 1. More generally those with d nodes occur in codimension d. In particular a d-dimensional linear family of curves should contain a finite number of such d-nodal curves. The classical problem - at least in the case of S being the projective plane - is to determine this number. The Gottsche conjecture states that the answer should be topological, given by a universal degree d polynomial in the four numbers C.C, c_1(S).C, c_1(S)^2 and c_2(S). This was proved recently by Yu-Jong Tzeng. I will explain a simpler proof which was joint work with Martijn Kool and Vivek Shende. The treatment will be very low-tech; I won't assume any prior knowledge. The main tool is Euler characteristics (which I will also explain).

Speaker:  Richard Thomas
Institution:  Imperial College


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