A compactification of the space of maps from curves (Q2862123)

The authors construct a new compactification \(\overline{\mathfrak{U}}_{g, \mu }(X,\beta )\) of the space of all maps \(f\) from genus \(g\), \(n\)-pointed nonsingular projective curves \((C,p_1,\ldots,p_n)\) to a nonsingular projective variety \(X\), representing class \(\beta \in A_1(X)\) such that \(f\) is ramified only possibly at \(p_1, \dots, p_n\), with the ramification indices \(\mu_1,\dots, \mu _n\in\mathbb Z\) respectively and \(f(p_i)\neq f(p_j)\) for \(i\neq j\).NEWLINENEWLINEThe boundary of \(\overline{\mathfrak{U}}_{g, \mu }(X,\beta )\) consists of maps from \(n\)-pointed prestable genus \(g\) curves to Fulton-MacPherson degeneration spaces of \(X\). The authors proved that \(\overline{\mathfrak{U}}_{g, \mu }(X,\beta )\) is a proper Deligne-Mumford stack equipped with a natural virtual fundamental class, which can be used to define the ramified Gromov-Witten invariants. \textit{R. Pandharipande} [Commun. Math. Phys. 208, No. 2, 489--506 (1999; Zbl 0953.14036); in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 503--512 (2002; Zbl 1047.14043)] conjectured that these ramified Gromov-Witten invariants are equal to BPS numbers.