Genus decreasing formula for higher genus Welschinger invariants
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Mathematische Zeitschrift
Genus decreasing formula for higher genus Welschinger invariants Yanqiao Ding1 Received: 2 April 2018 / Accepted: 22 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the changes experienced by higher genus Welschinger invariants of real del Pezzo surfaces in families undergoing a nodal degeneration and obtain a genus decreasing wallcrossing formula. Keywords Genus decreasing formula · Welschinger invariants · Real enumerative geometry Mathematics Subject Classification 14N10 · 14N35 · 53D45
1 Introduction Welschinger invariants can be considered as genus zero open Gromov–Witten invariants. Welschinger [32,33] first defined it by a signed counting of real rational pseudo-holomorphic curves which realize a given homology class and pass through a given real collection of points in a real rational symplectic 4-manifold. Del Pezzo surfaces are smooth rational surfaces with ample anti-canonical class −K . The degree of a del Pezzo surface X is K X2 . Any del Pezzo surface is isomorphic either to CP 1 × CP 1 or to CP 2 blown up at 0 ≤ k ≤ 8 generic points. A real structure τ X on a del Pezzo surface X is an anti-holomorphic involution. Denote by RX the fixed point set of τ X , and call RX the real part of the real del Pezzo surface X . The complex structure of real del Pezzo surfaces is enough generic to turn Welschinger’s count into a count of real algebraic rational curves. Actually, it follows from the fact that genus zero Gromov–Witten invariants GW0 (X , D) of a del Pezzo surface X are enumerative, i.e. on a generic X , GW0 (X , D) counts irreducible rational curves in a given divisor class D and passing through −D K X − 1 generic points in X (see [14, Theorem 4.1, Lemma 4.10] or [28, Sect. 10]). Vakil [31] observed that genus g ≥ 1 Gromov–Witten invariants GWg (X , D) of a del Pezzo surface X are also enumerative. Itenberg–Kharlamov–Shustin [24, Sect. 2] showed that Welschinger invariants as well as modified Welschinger invariants introduced in [20] are enumerative except for the case of del Pezzo surface X of degree 1 and D = −K X .
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Yanqiao Ding [email protected] School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
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Y. Ding
In this exceptional case, Itenberg–Kharlamov–Shustin [24, Sect. 2] showed that the number of solutions is still finite, but certain solutions may acquire some non-trivial multiplicity. In the algebro-geometric setting, Itenberg–Kharlamov–Shustin studied the positivity of Welschinger invariants and asymptotically logarithmic equivalence of Welschinger and Gromov–Witten invariants of del Pezzo surfaces in [16–22]. Their results not only showed the existence of real rational curves in a given divisor class and passing through generic real points of appropriate number in a real del Pezzo surface of degree ≥ 2, but also showed that the number of such curves are abundant. In the symplectic setting, Welschinger [32– 34] and Brugallé–Puignau [6,7] investigated the wal
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