Gromov-Witten invariants of blow-ups (Q2747851)

The author studies the behavior of the (genus zero) Gromov-Witten invariants of an arbitrary smooth projective variety \(X\) under blow-ups of some points. This task is motivated by several reasons, as the Gromov-Witten invariants of a blow-up variety \(\widetilde X\) might provide both interesting examples in general Gromov-Witten theory and explicit solutions to particular enumerative problems on the original variety \(X\) itself.NEWLINENEWLINE The first part of the author's work exhibits an efficient algorithm to compute the Gromov-Witten invariants of a blow-up \(\widetilde X\) in terms of those of a smooth variety \(X\). This is followed by an analysis of the enumerative significance of the Gromov-Witten invariants of the blow-up \(\widetilde X\) mainly with a view toward counting certain irreducible curves on \(X\) not contained in the exceptional divisor of the blow-up. In the second part of the paper, the author specializes his general results to blow-ups of a projective space \(\mathbb{P}^r\). It turns out that many Gromov-Witten invariants of these blow-ups can be interpreted as the number of rational curves in \(\mathbb{P}^r\) having prescribed global multiplicities in the blow-up points. This is enhanced by various concrete numerical examples given at the end, including an efficient method to compute multiple cover contribution factors for finite covers of infinitesimally rigid smooth rational curves on a Calabi-Yau threefold.NEWLINENEWLINE This work is part of the author's Ph.D. thesis written at the University of Hannover, Germany, with \textit{K. Hulek} as academic advisor [Gromov-Witten and degeneration invariants: Computation and enumerative significance (Hannover: Universität Hannover) (1998; Zbl 0902.14021)].