Irreducible degenerations of primary Kodaira surfaces

关于Fano簇退化的子簇

On Subvarieties of Degenerations of Fano Varieties

本文研究Fano簇及可分有理连通簇退化上的几何不可约子概型，证明其维数存在由一般纤维Fano指数给出的下界，拓展了相关子概型存在性结论的几何性质研究。

Real Kodaira surfaces

In this paper we give the topological classification of real primary Kodaira surfaces and we describe in detail the structure of the corresponding moduli space. One of the main tools is the orbifold fundamental group of a real variety. Our first result is that if $(S, σ)$ is a real primary Kodaira surface, then the differentiable type of the pair $(S,σ)$ is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of $(S, σ)$. Finally, we show that once we fix the topological type of $(S, σ)$ corresponding to a…

Cremona 等价与 log 罕见维数

Cremona equivalence and log Kodaira dimension

本文研究了Cremona等价与log Kodaira维度的关系，发现一个有理曲面在项目空间中具有负log Kodaira维度但不与平面Cremona等价，展示了log版本的Castelnuovo理性准则不成立。

Exceptional collections on Dolgachev surfaces associated with degenerations

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…

Degeneration of differentials and moduli of nodal curves on $K3$ surfaces

We consider, under suitable assumptions, the following situation: $\mathcal B$ is a component of the moduli space of polarized surfaces and $\mathcal V_{m,δ}$ is the universal Severi variety over $\mathcal B$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal B$ and $C\in |mH|$ irreducible with exactly $δ$ nodes as singularities. The moduli map $\mathcal V\to \mathcal M_g$ of an irreducible component $\mathcal V$ of $\mathcal V_{m,δ}$ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in $\mat…

旗簇线性退化中的正则性

Regularity in Linear Degenerations of Flag Variety

该研究针对旗簇线性退化的正则性问题，分类了光滑的（部分）旗簇线性退化，分析不可约退化奇点轨迹维数并引入表示空间新分层，最终证明不可约线性退化是正规簇。

Torsors on semistable curves and degenerations

In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible nodal curve} and the second one is the construction of a flat degeneration of the moduli space of semistable $G$-torsors when the smooth curve degenerates to an irreducible nodal curve. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an ap…

Nodal degenerations of plane curves and Galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois coverings.

Godeaux surfaces with an Enriques involution and some stable degenerations

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…

Initial degenerations of spinor varieties

We construct closed immersions from initial degenerations of the spinor variety $\mathbb{S}_n$ to inverse limits of strata associated to even $Δ$-matroids. As an application, we prove that these initial degenerations are smooth and irreducible for $n\leq 5$ and identify the log canonical model of the Chow quotient of $\mathbb{S}_5$ by the action of the diagonal torus of $\operatorname{GL}(5)$.

Betti numbers of real semistable degenerations via real logarithmic geometry

Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work…

Kodaira 维数为一的椭圆面模空间

Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

本文通过墙交叉技术和扭曲稳定映射技术，研究Kodaira 维数为一的椭圆面模空间，构造无限序列的不可约分量并描述其边界及稳定约简步骤。

Halphen pencils on quartic threefolds

For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.

Linear degenerations of flag varieties: partial flags, defining equations, and group actions

We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degener…

纤维化曲面上层的退化

Degenerations of Sheaves on Fibered Surfaces

本文利用扩展退化为纤维化曲面构造稳定层模叠并得到枚举不变量，证明粘合情形的退化公式，并计算某些椭圆曲面上稳定层模空间的椭圆亏格。

由旗簇上的簇结构产生的 Richardson 簇的半环面退化

Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties

本文证明旗簇上由簇结构产生的环面退化可诱导 Richardson 簇的半环面退化，并将其作为 Morier-Genoud 与 Kogan-Miller 相关构造的推广。

对数Fulton-MacPherson配置空间

Logarithmic Fulton--MacPherson configuration spaces

本文利用对数几何技术构建了Fulton-MacPherson配置空间的对数类比，并构造了其对数平滑退化，核心贡献是通过退化公式描述了退化后的不可约分量。

Normality of Orthogonal and Sympletic Nilpotent Orbit Closures in Positive Characteristic

In this note we investigate the normality of closures of orthogonal and symplectic nilpotent orbits in positive characteristic. We prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure, and conversely if the closure is normal, then any type e minimal irreducible degeneration does not occur in it. Here, the minimal irreducible degenerations of a nilpotent orbit are introduced by W. Hesselink in [6] (or see [11] from which we take Table 1 for the complete list of all minimal irreducible degenerati…

Degeneration and curves on K3 surfac

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces contain infinite number of irreducible rational curves. It is known that all K3 surfaces, except those contained in the countable union of hypersurfaces in the moduli space of K3 surfaces satisfy this property. In this paper we present a new approach for constructing curves on varieties which admit nice degenerations. We apply this technique to the above problem …

高效算法用于极大 matroid 衰减和电路品种的不可约分解

Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties

本文提出高效算法用于分解 matroid 的电路品种，通过识别极大衰减来实现显式且可计算的不可约分解。

走向Lang-Vojta猜想的退化研究

Towards Lang--Vojta via Degeneration

本文通过研究曲线模空间的算术和几何性质，证明了关于数字段上变种S-积分点有限性和Zariski退化的结果，并提供了几何上不可约除子的例子，回答了关于正常射影品种的积分点有限性问题。

Kodaira dimension of moduli of special cubic fourfolds

A special cubic fourfold is a smooth hypersurface of degree three and dimension four that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether-Lefschetz divisors C_d in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the "low-weight cusp form trick" of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of C_d. For example, if d = 6n + 2, then we show that C_d is of general type …

光滑拟射影簇之间的平坦态射的导出范畴上的半扭转

Half-spherical twists on derived categories of coherent sheaves

本文证明平坦态射纤维上的球状对象诱导纤维导出范畴的自等价，为椭圆曲面奇异纤维和K3曲面退化提供新对称性，并揭示与环面半扭转的联系。

曲线上向量丛模空间半稳定退化的几何

Geometry of a semistable degeneration of moduli of vector bundles on curves

研究具有一个节点的不可约曲线上稳定秩2度1向量丛的Gieseker模空间，通过两步爆破和自然对合粘合描述其正规化，计算了Picard群、虚拟Poincaré多项式及交集上同调的闭式公式。

Contractible curves on a rational surface

In this paper we prove that if S is a smooth, irreducible, projective, rational, complex surface and D an effective, connected, reduced divisor on S, then the pair (S,D) is contractible if the log-Kodaira dimension of the pair is $-\infty$. More generally, we even prove that this contraction is possible without blowing up an assigned cluster of points on S. Using the theory of peeling, we are also able to give some information in the case D is not connected.

Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injecti…

The monodromy property for K3 surfaces allowing a triple-point-free model

The aim of this thesis is to study under which conditions $K3$ surfaces allowing a triple-point-free model satisfy the monodromy property. This property is a quantitative relation between the geometry of the degeneration of a Calabi-Yau variety $X$ and the monodromy action on the cohomology of $X$: a Calabi-Yau variety $X$ satisfies the monodromy property if poles of the motivic zeta function $Z_{X,ω}(T)$ induce monodromy eigenvalues on the cohomology of $X$. In this thesis, we focus on $K3$ surfaces allowing a triple-point-free model, i.e., $K3$ surfaces allowing a strict normal crossings mod…

Convergence of polarizations, toric degenerations, and Newton-Okounkov bodies

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its assoc…

Chow-Kunneth decomposition for universal families over Picard modular surfaces

We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the work of Gordon, Hanamura and Murre we use Relatively Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular Surfaces in order to describe complete degenerations of families of abelian varieties. We furthermore prove vanishing results for cohomology groups of irreducible representations of certain arithmetic subgroups in SU(2,1) using the non-…

Real-normalized differentials: limits on stable curves

We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We determine all possible limits of RN differentials in degenerating sequences of smooth curves, and describe the limit in terms of solutions of the corresponding Kirchhoff problem. We further show that the limit of zeroes of RN differentials is the set of zeroes of a twisted meromorphic RN differential, which we explicitly construct. Our main new tool is an explicit solution of the jump problem on Riemann surfaces in plumbing coordinates, by using the Cauchy kernel on the normalizat…

Genus g Gromov-Witten invariants of Del Pezzo surfaces: Counting plane curves with fixed multiple points

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus $g$ curves in any divisor class $D$ on the blow-up of the plane at up to five points (no three collinear). We then show that these numbers give the genus $g$ Gromov-Witten invariants of the surface. Finally, we suggest a direction from which the remaining del Pezzo surfaces …

关于曲面退化中的Galois轨道界

Galois Orbit Bounds for Surface Degenerations

研究曲面退化中 Picard 等级跳跃的有限性，利用 Galois 轨道高度界和混合 Hodge 结构分析。