We analyze the local structure of moduli space of genus one stable quasimaps. Combining it with the p-fields theory developed in [8], we prove the hyperplane property for genus one invariants of stable quasimaps to hypersurface in P n .
By the reduced component in a moduli space of stable quasimaps to n-dimensional projective space P n we mean the closure of the locus in which the domain curves are smooth. As in the moduli space of stable maps, we prove the reduced component is smooth in genus 2, degree ě 3.Then we prove the virtual fundamental cycle of the moduli space of stable quasimaps to a complete intersection X in P n of genus 2, degree ě 3 is explicitly expressed in terms of the fundamental cycle of the reduced component of P n and virtual cycles of lower genus ă 2 moduli spaces of X.
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