Limit linear series and ranks of multiplication maps

Liu, Fu; Osserman, Brian; Teixidor i Bigas, Montserrat; Zhang, Naizhen

doi:10.1090/tran/8230

Limit linear series and ranks of multiplication maps
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by Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas and Naizhen Zhang PDF

Trans. Amer. Math. Soc. 374 (2021), 367-405 Request permission

Abstract:

We develop a new technique to study ranks of multiplication maps for linear series via limit linear series and degenerations to chains of elliptic curves. We prove an elementary criterion and apply it to proving cases of the Maximal Rank Conjecture. We give a new proof of the case of quadrics, and also treat several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails.

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