An effective algorithm of determining Gromov–Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound (1)) from Gromov–Witten invariants of the ambient space is proposed.

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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.

We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.