Asymptotically saturated toric algebras

Asymptotically maximal families of hypersurfaces in toric varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $Δ$ there are families of hypersurfaces with the Newton polytopes $(λΔ)_{λ\in \mathbb{N}}$ that are asymptotically maximal when $λ$ tends to infinity. We also show that these results ge…

环面点爆破的重数为1的Cox环

Multiplicity-One Cox Rings of Toric Point Blow-Ups

该研究针对环面点爆破的Cox环，给出Rees重数1生成的有限判据，将其应用于伪加权射影空间与射影环面曲面，得到重数1生成的等价条件。

一致K-稳定环面簇的渐近Chow稳定性

Asymptotic Chow stability of uniformly K-stable toric varieties

本文研究极化环面簇，在Futaki-Ono不变量消失的假设下，给出了一致K-稳定极化环面簇(X,L)渐近Chow多稳定的充分判据，并证明所有Futaki-Ono不变量消失的一致K-稳定光滑极化环面簇都是渐近Chow多稳定的。

Quasicoherent sheaves on toric schemes

Let X be the toric scheme over a ring R associated with a fan Sigma. It is shown that there are a group B, a B-graded R-algebra S and a graded ideal I of S such that there is an essentially surjective, exact functor ~ from the category of B-graded S-modules to the category of quasicoherent O_X-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection Xi_F from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-O_X-modules of ~F. If Sigma is simplicial, the above data can be chosen such that ~ vanishes precisely on I-t…