When are multidegrees positive?

热带多重度何时为正？

When are tropical multidegrees positive?

该论文研究热带多重度的正性，引入投影纯性与 facet 可选择性条件，推广 He 的定理，同时指出增广 Bergman 扇对应的热带体积多项式具有洛伦兹型性质。

On multidegrees of polynomial automorphisms of C^3

We prove that for every pair of positive integers a, b there exists a number c_0 such that for every c>=c_0 one can find a tame polynomial automorphism of C^3 with multidegree equal to (a,b,c).

多重次数、族与整相关

Multidegrees, families, and integral dependence

本文研究多重次数在族中的半连续性及整相关检测，提出 polar-Segre 重数并给出新的整相关准则，证明 Segre 数是关键不变量并推广相关上半连续性结果。

Covering gonalities of complete intersections in positive characteristic

We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli-de Poi-Ein-Lazarsfeld-Ullery for complex varieties. We show that over an arbitrary field a smooth multidegree $(d_1,\ldots,d_k)$ complete intersection in $\mathbb{P}^N$ has separable covering gonality at least $d-N+1$, where $d=d_1+\cdots+d_k$. We also show that the very general such variety has covering gonality at least $\frac{d-N+2}{2}$.

Everywhere local solubility for hypersurfaces in products of projective spaces

We prove that a positive proportion of hypersurfaces in products of projective spaces over $\mathbb{Q}$ are everywhere locally soluble, for almost all multidegrees and dimensions, as a generalization of a theorem of Poonen and Voloch. We also study the specific case of genus $1$ curves in $\mathbb{P}^1 \times \mathbb{P}^1$ defined over $\mathbb{Q}$, represented as bidegree $(2,2)$-forms, and show that the proportion of everywhere locally soluble such curves is approximately $87.4\%$. The proportion of these curves in $\mathbb{P}^1 \times \mathbb{P}^1$ soluble over $\mathbb{Q}_p$ is a rational …

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。