On compactifications of the Steinberg zero-fiber

Toric Elliptic Fibrations and F-Theory Compactifications

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibe…

精彩群紧化在Fano形变下的刚性

Rigidity of wonderful group compactifications under Fano deformations

本文证明了De Concini和Procesi构造的精彩群紧化在Fano形变下具有刚性，即族中一个纤维同构于该紧化则所有纤维均同构，从而回答了Bien和Brion提出的问题。

Hyperplane arrangements and mixed Hodge numbers of the Milnor fiber

For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…

$SU(n) \times \mathbb{Z}_2$ in F-theory on K3 surfaces without section as double covers of Halphen surfaces

We investigate F-theory models with a discrete $\mathbb{Z}_2$ gauge symmetry and $SU(n)$ gauge symmetries. We utilize a class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2, to yield genus-one fibered K3 surfaces with a bisection, but lacking a global section. We consider F-theory compactifications on these K3 surfaces times a K3 surface to build such models. We construct Halphen surfaces of index 2 with type $I_n$ fibers, and we take double covers of these surfaces to obtain K3 surfaces without a section with two type $I_n$ fibers, and K3 surfaces…

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

We prove that smooth Fano threefolds have toric Landau--Ginzburg models. More precise, we prove that their Landau--Ginzburg models, presented as Laurent polynomials, admit compactifications to families of K3 surfaces, and we describe their fibers over infinity. We also give an explicit construction of Landau--Ginzburg models for del Pezzo surfaces and any divisors on them.

算术 Satake 紧化与代数 Drinfeld 模形式

Arithmetic Satake compactifications and algebraic Drinfeld modular forms

本文构造任意秩 Drinfeld 模格式的算术 Satake 紧化并延拓泛族，据此定义代数 Drinfeld 模形式与 Hecke 作用，获得广义 Hasse 不变量，并在 Cohen-Macaulay 猜想下建立模 v 的 Jacquet-Langlands 对应。

On equivariant principal bundles over wonderful compactifications

Let $G$ be a simple algebraic group of adjoint type over $\mathbb C$, and let $M$ be the wonderful compactification of a symmetric space $G/H$. Take a $\widetilde G$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a complex reductive algebraic group and $\widetilde G$ is the universal cover of $G$. If the action of the isotropy group $\widetilde H$ on the fiber of $E$ at the identity coset is irreducible, then we prove that $E$ is polystable with respect to any polarization on $M$. Further, for wonderful compactification of the quotient of $\text{PSL}(n,{\mathbb C})$, $n\,\neq\, 4$…

Hodge numbers of Fano threefolds via Landau--Ginzburg models

For each smooth Fano threefold $X$ with Picard number 1 we consider a weak Landau--Ginzburg model, that is a fibration over $\mathbb C^1$ given by a certain Laurent polynomial. In the spirit of L. Katzarkov's program we prove that the number of irreducible components of the central fiber of its compactification is $h^{1,2}(X)+1$. In particular, it does not depend on the compactification. The question of dependence on the model is open; however we produce examples of different weak Landau--Ginzburg models for the same variety with the same number of components of the central fiber.

Poincaré duality of wonderful compactifications and tautological rings

Let $g \geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincaré duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications.

中心同源与约化群中的共轭类

Central isogenies and conjugacy classes in reductive groups

将Steinberg关于半单元素中心化子分量群的描述推广到解释当通用覆盖非平展时幂单元素中心化子非约化的现象，并应用于L-参数模空间和通用变形环的特殊纤维中的重数计算。

特征簇上通用几何空间的实谱紧化

Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties

本研究构造了SLₙ中有限生成群特征簇实谱紧化上的通用几何空间，证明投影映射可Γ等变延拓，其纤维同胚于实闭域上代数集的阿基米德谱，并计算了纤维在Berkovich解析化中的像。

Motivic Volumes of Fibers of Tropicalization

Let $T$ be an algebraic torus over an algebraically closed field, let $X$ be a smooth closed subvariety of a $T$-toric variety such that $U = X \cap T$ is not empty, and let $\mathscr{L}(X)$ be the arc scheme of $X$. We define a tropicalization map on $\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U)$, the set of arcs of $X$ that do not factor through $X \setminus U$. We show that each fiber of this tropicalization map is a constructible subset of $\mathscr{L}(X)$ and therefore has a motivic volume. We prove that if $U$ has a compactification with simple normal crossing boundary, then the g…

Linked Grassmannians and crude limit linear series

In math.AG/0407496, a new construction of limit linear series is presented which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The boundary of the compactification consists of crude limit series, and maps with positive-dimensional fibers to crude limit series of Eisenbud and Harris. In this paper, we carry out a careful analysis of the linked Grassmannian to obtain an upper bound on the dimension of the fibers of the map on crude limit series, thereby concluding an upper bound on the dimension of the locus of…

等残曲线

Isoresidual curves

本文研究等残纤维的结构，探讨其翻译结构的特征及欧拉示性，并分类零点数任意的零 genus 标层的连通分支。