论文标题
Gröbner变性，司法和通用集群代数的家庭
Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
论文作者
论文摘要
让$ v $是由加权均质的$ j $和$ c $的加权投影品种，在$ j $带$ m $ rays的Gröbner粉丝中，最大圆锥。我们在$ \ Mathbb a^m $上构建了一个平坦的家族，该家庭组装了与$ c $的所有面孔相关的$ v $的gröbner变性。这是与重量相关的经典单参数Gröbner变性的多参数概括。我们解释了如何从卡夫·曼农（Kaveh-Manon）最近的关于福利品种的福利家族分类的工作来建立我们的家庭：它是由rees algebra定义的带有基础$ x_c $（与$ c $相关的co $）沿环球torsor $ \ mathbb $ \ nathbb a^m a^m a^m \ x_ x_ x_ x_ x_ x_ x_c $ \ x_c $ \ x_c $的the thuck fors的折叠。我们将此结构应用于Grassmannians $ {\ rm gr}（2，\ Mathbb c^n）$，其plücker嵌入式和Grassmannian $ {\ rm gr} \ big（3，\ Mathbb c^6 \ big）$带有群集的组合。在每种情况下，都存在一个独特的最大Gröbner锥，其初始理想是群集复合体的Stanley-Reisner理想。我们表明，具有通用系数的相应群集代数是定义与该锥体相关的平坦家族的代数。此外，对于$ {\ rm gr}（2，\ mathbb c^n）$，我们展示了如何将Escobar-Harada的Newton-Okounkov身体突变作为热带化集群突变恢复。
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gröbner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $\mathbb A^m \to X_C$. We apply this construction to the Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Plücker embeddings and the Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for ${\rm Gr}(2,\mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.
