scholarly journals Towards an Enumerative Geometry of the Moduli Space of Curves

1983 ◽  
pp. 271-328 ◽  
Author(s):  
David Mumford
Keyword(s):  
Moduli Space ◽  
Enumerative Geometry ◽  
Moduli Space Of Curves

The Kodaira dimension of the moduli space of curves of genus ?23

1987 ◽  
Vol 90 (2) ◽  
pp. 359-387 ◽  
Author(s):  
David Eisenbud ◽  
Joe Harris
Keyword(s):  
Moduli Space ◽  
Kodaira Dimension ◽  
Moduli Space Of Curves

ON GROMOV–WITTEN INVARIANTS OF DEL PEZZO SURFACES

2013 ◽  
Vol 24 (07) ◽  
pp. 1350054 ◽  
Author(s):  
MENDY SHOVAL ◽  
EUGENII SHUSTIN
Keyword(s):  
Quantum Cohomology ◽  
Enumerative Geometry ◽  
Del Pezzo Surfaces ◽  
Manuscripta Math ◽  
Moduli Space Of Curves ◽  
Divisor Class ◽  
Genus Zero ◽  
Type Formula ◽  
Del Pezzo ◽  
Generic Points

We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on [Formula: see text], the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type [Formula: see text], the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of [Formula: see text] via enumerative invariants of [Formula: see text], using Vakil's extension of the Abramovich–Bertram formula.

scholarly journals Brill–Noether loci in codimension two

2013 ◽  
Vol 149 (9) ◽  
pp. 1535-1568 ◽  
Author(s):  
Nicola Tarasca
Keyword(s):  
Moduli Space ◽  
Moduli Space Of Curves ◽  
Stable Curves ◽  
Codimension Two

AbstractLet us consider the locus in the moduli space of curves of genus$2k$defined by curves with a pencil of degree$k$. Since the Brill–Noether number is equal to$- 2$, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

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