C. A. M. Peters, J. H. M. Steenbrink
We prove that the Leray spectral sequence in rational cohomology for the quotient map $U_{n,d} \to U_{n,d}/G$ where $U_{n,d}$ is the affine variety of equations for smooth hypersurfaces of degree $d$ in $\PP^n(\C)$ and $G$ is the general linear group, degenerates at $E_2$.
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Let Z be an arrangement of submanifolds in a complex compact algebraic manifold X. We allow some kind of singular intersections. We consider the Leray spectral sequence of the embedding of the U=X-Z into X and formulate a condition sufficient for degeneration of this spectral sequence on E_3-table.
We show that the moduli space of stable n-pointed rational curves can be flatly degenerated to a projective toric variety. We arrive at this by showing that the Chow quotients of the Grassmannians admit toric degenerations, which in turn, follows from a theorem that we prove for toric degenerations of more general Chow quotients. Along the way, we also argued that GIT quotient of a flat family is again flat. In particular, all GIT quotients of flag varieties by maximal tori can be flatly degenerated to projective toric varieties ([9] and [12]).
Let $Q^*$ denote the dual of the quotient bundle on the Grassmannian $G(2,n)$. We prove that the ideal of $Q^*$ in its natural embedding has initial ideal equal to the Stanley-Reisner ideal of a certain unobstructed simplicial complex. Furthermore, we show that the coordinate ring of $Q^*$ has no infinitesimal deformations for $n>5$.
Broken Toric Varieties and Cell-Compatible Sheaves
本文利用常值层到多面体复形的导出前推研究破碎环面簇的上同调，证明Deligne型分解定理和谱序列退化，并给出部分情形的Betti数显式公式。
We consider a surface that admits a $\mathbb{Q}$-Gorenstein degeneration to a cyclic quotient singularity $\frac{1}{dn^2}(1,dna-1)$. Under several technical assumptions, we construct $d$ exceptional vector bundles of rank $n$ which are orthogonal to each other.
We give an explicit description of the Godeaux surfaces that admit an involution such that the quotient surface is birational to an Enriques surface; these surfaces give a 6-dimensional unirational irreducible subset of the moduli space of surfaces of general type. In addition, we describe the Enriques surfaces that are birational to the quotient of a Godeaux surface by an involution and we show that they give a 5-dimensional unirational irreducible subset of the moduli space of Enriques surfaces. Finally, by degenerating our description we obtain some examples of non-normal stable Godeaux sur…
Degenerations of cluster type varieties
本文研究了簇型品种和表面的退化，证明了某些条件下退化是有限商，并对加权射影平面的退化进行了分类。
In this paper we calculate fundamental groups (and some of their quotients) of complements of four toric varieties branch curves. For these calculations, we study properties and degenerations of these toric varieties and the braid monodromies of the branch curves in $\mathbb{CP}^2$. The fundamental groups related to the first three toric varieties turn to be quotients of the Artin braid groups $\mathcal{B}_5$, $\mathcal{B}_6$, and $\mathcal{B}_4$, while the fourth one is a certain quotient of the group $\tilde{\mathcal{B}}_6 = \mathcal{B}_6/<[X,Y]>$, where $X, Y$ are transversal. The quotients…
Let X be a surface with quotient singularities which admits a smoothing to the plane. We prove that X is a deformation of a weighted projective plane P(a^2,b^2,c^2), where a,b,c is a solution of the Markov equation a^2+b^2+c^2=3abc. We also prove a generalisation for del Pezzo surfaces of degree K^2 at least 5.
We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.
Degenerations of twisted maps to algebraic stacks
本文定义了具有射影好模空间的商栈的扭曲映射，证明所得函子满足真的赋值判据的存在性部分。
The Mirror Clemens-Schmid Sequence
本文提出联系光滑簇与补开集上同调的四项长正合序列，证明其兼容反常 Leray 滤过和混合 Hodge 结构，并通过 K3、Calabi–Yau 及五次三维簇退化验证其为 Clemens–Schmid 序列的镜像。
Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence of the map from $V_{n,d}$ to the homotopy quotient $(V_{n,d})_{hG'}$ degenerates at $E_2$, and so does the Leray spectral sequence of the quotient map $V_{n,d}\to V_{n,d}/G'$ provided the geometric quotient $V…
An embedded variety is said to be well-poised when the associated initial ideal degenerations coming from points of the tropical variety are reduced and irreducible. Varieties with a well-poised embedding admit a large collection of explicitly constructible Newton-Okounkov bodies. This paper aims to study the well-poised property under torus quotients. Our first result states that GIT quotients of normal well-poised varieties by quasi-tori also have well-poised embeddings. As an application, we show that several Hassett spaces, $\overline{M}_{0,β}$, are well-poised under Alexeev's embedding. C…
Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev…
$\mathfrak{G}$-Quotients of Grassmannians and Equations
本文研究了Grassmannian的$\mathfrak{G}$-quotient结构，通过分析子torus作用的定义理想和退化，获得了一个双有理映射及其相关的新结果。
Let f be a degeneration of Kahler manifolds. The local invariant cycle theorem states that for a smooth fiber of the degeneration, any cohomology class, invariant under the monodromy action, rises from a global cohomology class. Instead of the classical cohomology, one may consider the non-abelian cohomology. This note demonstrates that the analogous non-abelian version of the local invariant cycle theorem does not hold if the first non-abelian cohomology is the moduli space (universal categorical quotient) of the representations of the fundamental group.
Initial degenerations of flag varieties
本文研究旗变种的初始退化，证明其在特定条件下光滑且不可约，并探讨了Chow商与log canonical模型的关系。
Degeneration of K3 surfaces with non-symplectic automorphisms
本文证明在特定条件下，具有非辛自同构的K3曲面不会退化，关键依赖于自同构在l-adic上同调权重过滤阶商上的有理作用。
We prove that the open subvariety $\operatorname{Gr}_0(3,8)$ of the Grassmannian $\operatorname{Gr}(3,8)$ determined by the nonvanishing of all Plücker coordinates is schön, i.e., all of its initial degenerations are smooth. Furthermore, we find an initial degeneration that has two connected components, and show that the remaining initial degenerations, up to symmetry, are irreducible. As an application, we prove that the Chow quotient of $\operatorname{Gr}(3,8)$ by the diagonal torus of $\operatorname{PGL}(8)$ is the log canonical compactification of the moduli space of $8$ lines in $\mathbb{…
We prove that if a symplectic 4-manifold $X$ becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold $X$ is originally rational. That is, a symplectic rational blow-up of a rational symplectic $4$-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of [Bădescu, 1986] in algebraic geometry under a mild additional condition.
The birational geometry of Markov numbers
本文利用双有理几何刻画复射影平面商奇点退化在Markov解间的连接路径，将其对应于循环商奇点翻转泛族的Mori train，并得到Markov数的新组合数据及与Frobenius唯一性猜想的联系。
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