Counting curves on rational surfaces |
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Authors: | Ravi Vakil |
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Institution: | (1) M.I.T. Department of Mathematics, 77 Massachusetts Ave. Rm. 2-271, Cambridge, MA 02139, USA. e-mail: vakil@math.mit.edu, US |
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Abstract: | In CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps,
and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count
irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus
Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree
at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus
Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface.
Received: 30 June 1999 / Revised version: 1 January 2000 |
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Keywords: | Mathematics Subject Classification (1991):14N10 14H10 14J10 |
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