We develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus-1 curves. We use this approach to prove a purely elementary criterion for proving cases of the Maximal Rank Conjecture, and then apply the criterion to several ranges of cases, giving a new proof of the case of quadrics, and also treating 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.

中文翻译：

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限制乘法映射的线性级数和秩

我们开发了一种新技术，通过限制线性级数和退化到属 1 曲线链来研究线性级数的乘法映射的秩。我们使用这种方法来证明证明最大秩猜想的情况的纯基本标准，然后将该标准应用于几个范围的情况，给出二次方程的情况的新证明，并且在三次方程的情况下也处理了几个族. 我们的证明不需要对接近方向的限制，因此我们在最大秩条件失败的曲线的模空间中恢复轨迹的新信息。