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Decomposition of degenerate Gromov-Witten invariants

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arxiv 1709.09864 v4 pith:S7JDYR2U submitted 2017-09-28 math.AG

Decomposition of degenerate Gromov-Witten invariants

classification math.AG
keywords decompositionfamilyfibregromov-witteninvariantslogarithmicaspectbeta
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family X->B with singular fibre over b_0 \in B yields a family M(X/B,\beta) -> B of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over b_0 in terms of rigid tropical curves. This generalizes one aspect of known results in the case that the fibre X_{b_0} is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

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  1. The integral Chow ring of $\mathscr{M}_{0}(\mathbb{P}^r, 2)$

    math.AG 2026-04 unverdicted novelty 6.0

    The integral Chow ring of M_0(P^r, 2) is presented as a quotient of a three-variable polynomial ring with all non-trivial relations encoded by two rational generating functions.