Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure
type: Item
instance of: scholarly article
title: Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure
zbMATH Open document ID: 1406.53088
author: Indranil Biswas
author: Ritwik Mukherjee
author: Varun Thakre
published in: Geometriae Dedicata
publication date: 2018-08-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.04900
review text: The number of irreducible, reduced, nodal, degree-\(d\) rational curves passing through the right number of general points in \(\mathbb C \mathbb P^2\) satisfies a nice recursion relation found by \textit{M. Kontsevich} and \textit{Yu. Manin} [AMS/IP Stud. Adv. Math. 1, 607--653 (1997; Zbl 0931.14030)] and also by \textit{Y. Ruan} and \textit{G. Tian} [Math. Res. Lett. 1, No. 2, 269--278 (1994; Zbl 0860.58006)]. This formula was generalized independently by \textit{E. Ionel} [Duke Math. J. 94, No. 2, 279--324 (1998; Zbl 0974.14038)] and \textit{R. Pandharipande} [Proc. Am. Math. Soc. 125, No. 12, 3471--3479 (1997; Zbl 0883.14031)] to the case of elliptic curves with fixed \(j\)-invariant, and by \textit{S. Katz} et al. [J. Algebr. Geom. 7, No. 3, 569--587 (1998; Zbl 0946.14031)] and \textit{A. Zinger} [J. Algebr. Geom. 13, No. 3, 547--561 (2004; Zbl 1063.14038)] to the case of genus-two curves with the fixed complex structure. The authors of the paper under review extended the result for the elliptic curves to del-Pezzo surfaces [\textit{I. Biswas} et al., C. R., Math., Acad. Sci. Paris 354, No. 5, 517--521 (2016; Zbl 1387.14099)]. In this paper they are extending the result for genus-two to del-Pezzo surfaces. To state their result, let \(X\) be a del-Pezzo surface with the second Betti number \(b_2\), \(\beta \in H_2(X,\mathbb Z)\), \(n_{0,\beta}\) the number of rational curves on \(X\) passing through the right number of points, \(\delta_\beta=-K_X\cdot \beta-1\). Let \(A\) be half the number of holomorphic automorphisms of a genus-two Riemann surfaces with a fixed complex structure. Then, under certain conditions to ensure genus-two regularity of the complex structures, \(A\) times the number of genus-two curves with this complex structure passing through a right number of points is \[ n_{0,\beta}\Big((2+b_2)\beta^2-10c_2(X)-K_X^2-12K_X^2/(K_X\cdot \beta) \] \[ +\sum_{\beta_1,\beta_2} \binom{\delta_\beta-1}{\delta_{\beta_1}}n_{0,\beta_1}n_{0,\beta_2}(\beta_1\cdo…
reviewed by: Amin Gholampour
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 6924059
zbMATH Keywords: genus two curves
zbMATH Keywords: del-Pezzo surface
zbMATH Keywords: symplectic invariant
zbMATH Keywords: enumerative invariant
MaRDI profile type: Publication
OpenAlex ID: W2766789975
arXiv ID: 1511.04900
cites work: Rational cuspidal curves on del-Pezzo surfaces
cites work: Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure
cites work: The quantum cohomology of blow-ups of \(\mathbb{P}^2\) and enumerative geometry
cites work: Q4846449
cites work: Genus \(1\) enumerative invariants in \(\mathbb{P}^n\) with fixed \(j\) invariant
cites work: Q4267949
cites work: Counting tropical elliptic plane curves with fixed \(j\)-invariant
cites work: Characteristic numbers of rational curves with cusp or prescribed triple contact
cites work: Q4232457
cites work: Enumerative geometry of elliptic curves on toric surfaces
cites work: Q4830110
cites work: Counting elliptic plane curves with fixed $j$-invariant
cites work: A mathematical theory of quantum cohomology
cites work: Enumeration of genus-two curves with a fixed complex structure in \(\mathbb{P}^2\) and \(\mathbb{P}^3\)
cites work: Completion of Katz-Qin-Ruan's enumeration of genus-two plane curves
cites work: Enumerative vs. symplectic invariants and obstruction bundles
DOI: 10.1007/S10711-017-0297-Y
description: scientific article