Nodal curves old and new
59 mins 41 secs,
248.65 MB,
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Description: |
Thomas, RPW (Imperial)
Friday 11 March 2011, 15:15-16:15 |
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Created: | 2011-03-18 10:07 | ||||
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Collection: | Moduli Spaces | ||||
Publisher: | Isaac Newton Institute | ||||
Copyright: | Thomas, RPW | ||||
Language: | eng (English) | ||||
Distribution: | World (downloadable) | ||||
Credits: |
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Explicit content: | No | ||||
Aspect Ratio: | 16:9 | ||||
Screencast: | No | ||||
Bumper: | UCS Default | ||||
Trailer: | UCS Default |
Abstract: | I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques, but techniques that one would never really 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 Göttsche 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). There are now proofs in various settings; a completely algebraic proof was found recently by Tzeng. I will explain a simpler approach which is joint work with Martijn Kool and Vivek Shende. |
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