Valerie Hower
We present a counterexample to the conjecture of Bihan, Franz, McCrory, and van Hamel concerning the maximality of toric varieties. There exists a six dimensional projective toric variety X with the sum of the mod 2 Betti numbers of X(R) strictly less than the sum of the mod 2 Betti numbers of X(C).
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We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) projectively normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric varieties as well as algorithms checking properties (i) and (ii) and further potential properties, in Particular a weaker version of (ii) asking for scheme-theoretic definition in degree 2.
We construct a 3-dimensional complete intersection toric variety on which the subadditivity formula doesn't hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2-dimensional normal toric varieties is also provided.
A toric Calabi-Yau counterexample to Schimpf's connected stable pairs pole conjecture
本文构造了一个光滑拟射影环面 Calabi-Yau 三维流形反例，其中本原类 \\(\beta=2[C_1]+[C_2]\\) 的连通 descendant 系数在 \\(p=1\\) 处出现非零极点，违反了 Schimpf 连通稳定对极点猜想的限制条件。
We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.
We give examples for existence and non-existence of categorical quotients for algebraic group actions in the categories of algebraic varieties and prevarieties. All our examples are subtorus actions on toric varieties.
Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement
该研究构造了所有维度d≥3的佐藤弱F等价猜想反例，提出戈伦斯坦弱F等价及对应改进猜想，证明其在d≤3及反例维度成立，还揭示其与自反单形包含连通性的关联。
The article has been withdrawn by the author due to the existence of counterexamples.
This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in projective four-space provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces.
The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties) in his celebrated book "Topological Methods in Algebraic Geometry". The generalization for singular varieties is due to Brasselet-Schürmann-Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are mentioned. After this review part, we…
A one-step counterexample to the normalized Nash blowup conjecture
本文构造了一个五维的正常奇异仿射交错簇，其归一化纳什吹胀已包含与原簇同构的仿射开子簇，从而在五维及以上所有特征下给出单步反例。
Surface counterexamples to the Eisenbud-Goto conjecture
本文构造了射影曲面在P^4中的反例，研究了其射影不变量、上同调性质和几何性质，推翻了Eisenbud-Goto猜想。
We present a smooth, complete toric threefold with no nontrivial nef line bundles. This is a counterexample to a recent conjecture of Fujino.
Codimension two torus actions on the affine space
本文分类了具有复杂度二环面作用、唯一不动点且商为环面爆破的光滑可缩仿射簇，为仿射几何线性化猜想提供潜在反例。
King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7.
On a Computational Approach to the Nash Blowup Problem
本文通过计算方法发现纳什吹胀猜想的反例，并提供了无法通过标准化纳什吹胀解析的典范簇例子，同时展示了大量计算证据支持猜想的剩余开放案例。
The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $\ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID…
A Resolution of the Diagonal for Smooth Projective Toric Varieties
研究光滑射影环面簇在主除子格条件不成立时的对角线分解问题，通过阶梯函数标记顶点并施加扇形对称假设，得到\(X_\Sigma\times X_\Sigma\)上\(\mathcal{O}_\Delta\)的局部自由分解，还探讨了变形参数\(\epsilon\)对分解族的影响。
Invariant Rings of $ \mathbb{G}_{a} $-Representations are not always Finitely Generated in Positive Characteristic
本文针对正特征域上 Weitzenböck 猜想，利用 Mukai 策略构造一个六维 $\mathbb{G}_{a}$-表示，并借助环面簇几何证明其不变环不是有限生成的，从而给出该猜想的反例。
We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of delta-vectors of reflexive polytopes.
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