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 $X$.
Categories: math.AG math.SG