关于 $F$-阈值相对于吹脱纤维的性质以及 $F$-理性

Mori Dream Spaces & Blow-Ups

Mori Dream Spaces and Blow-Ups

本文的目的是对Mori梦空间的一般理论进行综述，并特别关注这样一个问题：什么时候环面簇在一般点上的爆破是Mori梦空间？我们把Picard数为1的复曲面的问题转化为涉及射影平面上的点的插值问题。冈萨雷斯-卡鲁定理就是这样一个插值问题的例子，它给出了加权射影平面的新例子，这些平面在一般点处的爆破不是森梦空间。

$\mathbb{P}^{n-3}$在$n$点的爆破和旋量簇

Blow-ups of $\mathbb{P}^{n-3}$ at $n$ points and spinor varieties

Dolgachev和Castravet-Tevelev的工作建立了$\mathfrak{so}_{2n}$的半旋量表示的$2^{n-1}$权与通过在$n$点上炸毁$\mathbb{P}^{n-3}$而获得的变种$X_n$的Cox环的生成元之间的双射。通过将${\rm Cox}(X_n)$嵌入到偶旋量簇（偶半旋量表示的齐性空间）中，我们得到了这种双射的几何解释。通过偶旋量簇的相交环面平移几何地恢复了爆破$X_n$的Cox环。这些是Derenthal和Serganova-Skorobogatov在del Pezzo曲面上的结果的高维推广。

Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases. We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,...,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14t…

多重射影空间的爆破的Cox环

Cox rings of blow-ups of multiprojective spaces

该研究聚焦多重射影空间爆破的Cox环问题，刻画了不同参数下爆破空间的Mori锥，证明特定爆破空间为对数Fano簇并给出其Cox环显式表示，丰富了代数几何相关结构的研究。

Cohomological Blow Ups of Graded Artinian Gorenstein Algebras Along Surjective Maps

We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersecti…