Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces

Details Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces Gromov-Witten theory of root gerbes I: structure of genus $0$ moduli spaces

2009-07-13

arxiv.org/abs/0907.2087v2

Mathematics

Algebraic Geometry

v1: 43 pages; v2: revised version with new title, 37 pages

Scientific paper

Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r>1$ one defines the stack $\sqrt[r]{\mathcal{L}/X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$. Our main results are explicit constructions of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L}/X}$ starting from moduli stack of genus $0$ stable maps to $X$. As a consequence, we prove an exact formula expressing genus $0$ Gromov-Witten invariants of $\sqrt[r]{\mathcal{L}/X}$ in terms of those of $X$.

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