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SUMMARY:Nodal curves old and new - Thomas\, RPW (Imperial)
DTSTART:20110311T151500Z
DTEND:20110311T161500Z
UID:TALK30202@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:I will describe a classical problem going back to 1848 (Steine
 r\, Cayley\, Salmon\,...) and a solution using simple techniques\, but tec
 hniques that one would never really have thought of without ideas coming f
 rom string theory (Gromov-Witten invariants\, BPS states) and modern geome
 try (the Maulik-Nekrasov-Okounkov-Pandharipande conjecture). In generic fa
 milies of curves C on a complex surface S\, nodal curves -- those with the
  simplest possible singularities -- appear in codimension 1. More generall
 y 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 cu
 rves. The classical problem -- at least in the case of S being the project
 ive plane -- is to determine this number. The Gttsche conjecture states th
 at the answer should be topological\, given by a universal degree d polyno
 mial in the four numbers C.C\, c_1(S).C\, c_1(S)^2 and c_2(S). There are n
 ow proofs in various settings\; a completely algebraic proof was found rec
 ently by Tzeng. I will explain a simpler approach which is joint work with
  Martijn Kool and Vivek Shende.\n
LOCATION:Seminar Room 1\, Newton Institute
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