We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness criterion of Hironaka. The results extend to more general varieties defined locally by binomial equations.
Let $(X, A)$ be a polarized nonsingular toric 3-fold with not effective $A+K_X$. Then for any ample line bundle $L$ on $X$ the image of the embedding by the complete linear system of $L$ is an intersections of quadrics.
We study the Hadamard product of two varieties $V$ and $W$, with particular attention to the situation when one or both of $V$ and $W$ is a binomial variety. The main result of this paper shows that when $V$ and $W$ are both binomial varieties, and the binomials that define $V$ and $W$ have the same binomial exponents, then the defining equations of $V \star W$ can be computed explicitly and directly from the defining equations of $V$ and $W$. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result…
Let $X$ be any variety in characteristic zero. Let $V \subset X$ be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of $X$ except for V. It is a morphism $f: Y \to X$ , which does not modify the subset $ V $ and transforms $X$ into a toroidal embedding $Y$, with singularities extending those on $V$. Moreover, the exceptional divisor has simple normal crossings on $Y$. The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus $V$ and transforms $X$ into a nonsingular variety $Y$. Th…
We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the comp…
On Calabi-Yau Complete Intersections in Toric Varieties
研究了$d$维环面Fano簇中$r$半样本因子的Calabi-Yau完全交$V$的Hodge理论性质.我们的主要目的是证明第二作者提出的组合对偶与镜像对称预言的Hodge数的对偶是一致的。预计对任意维度奇异卡-丘变体$V$的镜像对称预测的完整验证需要考虑所谓的{ \em弦理论霍奇数} $h^{p,q}_{\rm st}(V)$。我们仅限于弦理论霍奇数$h^{0,q}_{\rm st}(V)$和$h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $ h^{0，q}（\widehat{V}）$ and $ h^{1，q}（\widehat{V}）$ of a $ MPCP $-desingularization $ \widehat{V} $ of $ V$。
