基于形式化去奇异化的超曲面伴随计算

利用环面 Artin 叠对二项式簇的去奇异化

Desingularization of binomial varieties using toric Artin stacks

本文利用 fantastacks 概念对代数闭域上的二项式簇进行有效去奇异化，大幅减少步骤和图表数，并扩展到多个二项式超曲面的部分同时正规交叉去奇异化。

环面作用、加权吹胀与平面曲线的脱 singularization

Torus actions, weighted blow-ups, and desingularization of plane curves

本文通过加权吹胀构造平面曲线的脱 singularization，利用环面作用和 equivariant 中心进行归纳论证，无需限制于完美基域。

On Green's proof of infinitesimal Torelli theorem for hypersurfaces

We prove an equivalence between the infinitesimal Torelli theorem for top forms on a hypersurface contained inside a Grassmannian $\mathbb G$ and the theory of adjoint volume forms presented in L. Rizzi, F. Zucconi, "Generalized adjoint forms on algebraic varieties", Ann. Mat. Pura e Applicata, in press. More precisely, via this theory and a suitable generalization of Macaulay's theorem we show that the differential of the period map vanishes on an infinitesimal deformation if and only if certain explicitly given twisted volume forms go in the generalized Jacobi ideal of $X$ via the cup produc…

多胞体（Polypols）的伴随形式与典范形式

Adjoints and Canonical Forms of Polypols

该研究从代数几何角度研究具有唯一伴随超曲面的平面与三维多胞体，关联正几何概念，解决Wachspress猜想部分情形，刻画凸多边形伴随曲线实拓扑，还探讨了与代数统计学的联系。

Projective geometry of Wachspress coordinates

We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform p…

对偶多面体的伴随：行列式表示与光滑性

Adjoints of Polytopes: Determinantal Representations and Smoothness

本文研究了多面体伴随超曲面的行列式表示，证明了所有多边形的伴随多项式可表示为三对角矩阵的行列式，并探讨了伴随超曲面的光滑性与高维表示问题。