On the Noether and the Cayley-Bacharach theorems with PD multiplicities

Mixed multiplicities of Divisorial Filtrations

Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m…

多重分次代数与多重分次线性系

Multigraded algebras and multigraded linear series

本文研究多重分次代数与多重分次线性系，定义体积函数并建立其与整体牛顿-奥昆科夫体纤维体积的关联，证明可分解分次下体积函数为非负系数多项式，定义混合重数并刻画其正性，统一了混合重数领域的相关结果。