Yonghwa Cho
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.
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We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.
In this paper we mainly describe $\mathbb{Q}$-Gorenstein smoothings of projective surfaces with only Wahl singularities which have birational fibers. For instance, these degenerations appear in normal degenerations of the projective plane, and in boundary divisors of the KSBA compactification of the moduli space of surfaces of general type [KSB88]. We give an explicit description of them as smooth deformations plus 3-fold birational operations, through the flips and divisorial contractions in [HTU13]. We interpret the continuous part (smooth deformations) as degenerations of certain curves in …
We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $ρ(X)+2$. Degenerations with $ρ(X)+2$ and $ρ(X)+1$ non-Du Val points are investigated.
We explicitly construct a $V$-normal crossing Gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toric geometry. Using this model, we calculate the stringy $E$-polynomial of the relative symmetric product. We also construct a minimal model of degeneration of Hilbert schemes explicitly.
We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polyto…
Normal stable degenerations of Noether-Horikawa surfaces
本文研究Noether-Horikawa曲面的正规稳定退化，分类了仅含$\mathbb{Q}$-Gorenstein可光滑化对数典范奇点的曲面，并给出整体可光滑化判据及模空间边界层描述。
A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with $p_g = 0$ and the boundary of the moduli space of deformations of the surface. In the present paper we analyze this relation for Godeaux surfaces. We provide a description of certain boundary components of the moduli space of Godeaux surfaces. Also we explicitly construct certain exceptional vector bundles of rank 2 on Godeaux surfaces, stable with respect to the canonical class, and examine the correspondence between the boundary components and such exceptional vector bundles…
We show the existence of a full exceptional collection in the graded stable derived category of a Gorenstein isolated quotient singularity using a result of Orlov (arXiv:math/0503632). We also show that the equivariant graded stable derived category of a Gorenstein Veronese subring of a polynomial ring with respect to an action of a finite group has a full strong exceptional collection, even if the corresponding quotient singularity is neither isolated nor Gorenstein.
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…
In this paper we consider Gorenstein stable surfaces with $K^2_X=1$ and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with $K_X^2=1$; for $p_g=2$ this leads to a rough stratification of the moduli space. Explicit non-Gorenstein examples show that we need further techniques to understand all possible degenerations.
In 1981 J. Wahl described smoothings of surface quotient singularities with no vanishing cycles. Given a smoothing of a projective surface X of this type, we construct an associated exceptional vector bundle on the nearby fiber Y in the case H^{2,0}(Y)=H^1(Y)=0. If Y is the projective plane we show that our construction establishes a bijective correspondence between the possible degenerate surfaces X and exceptional bundles on Y modulo dualizing and tensoring by line bundles. If Y is of general type then our construction establishes a connection between components of the boundary of the moduli…
In this note, we describe the possible singularities on a stable surface which is in the boundary of the moduli space of surfaces isogenous to a product. Then we use the $\mathbb Q$-Gorenstein deformation theory to get some connected components of the moduli space of stable surfaces.
Weighted projective degenerations of $\mathbb{P}^{n}$
研究射影空间 $\mathbb{P}^{n}$ 的加权射影 klt $\mathbb{Q}$-Gorenstein 退化，构造了任意维度的无穷多个新退化，并详细研究了任意加权射影三维簇的形变。
We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie group $\mathbb{G}_2$. It occurs that it is also the image of $\mathbb{P}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi--Yau threefolds. We …
Inspired by Bondal's conjecture, we study the behavior of exceptional sequences of line bundles on rational C*-surfaces under homogeneous degenerations. In particular, we provide a sufficient criterion for such a sequence to remain exceptional under a given degeneration. We apply our results to show that, for toric surfaces of Picard rank 3 or 4, all full exceptional sequences of line bundles may be constructed via augmentation. We also discuss how our techniques may be used to construct noncommutative deformations of derived categories.
We prove the existence of a full exceptional collection for the derived category of equivariant matrix factorizations of an invertible polynomial with its maximal symmetry group. This proves a conjecture of Hirano--Ouchi. In the Gorenstein case, we also prove a stronger version of this conjecture due to Takahashi. Namely, that the full exceptional collection is strong.
The degeneration of a family of rational surface automorphisms
研究有理表面族的自同构退化，发现极限映射在特殊纤维上为恒等映射，通过吹开后得到诱导的双有理映射，其动力学度数为16。
We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear…
Categorical genus for log del Pezzo surfaces with cyclic quotient singularities
本文计算具有循环商奇点的log del Pezzo曲面的典范栈的范畴亏格，通过Serre对偶和惯性Riemann–Roch得到闭式公式，并证明其与Tveiten可变亏格及toric退化下的Laurent多项式纤维亏格一致。
Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with respect to its Newton diagram. Let $π\colon Y\to X$ be a resolution. We show that there are not more than 2 non-rational divisors $E_i$, $i=1,2$, on $Y$ such that $π(E_i)=o$ and discrepancy $a(E_i,X)\leq 1$. When such divisors exist, we describe them as exceptional divisors of certain blowups of $X$ and study their birational type.
We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.
We describe here a degeneration of the symplectic desingularization of the moduli spaces of topologically trivial $GL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$-Higgs bundles over a hyperelliptic curve, into O'Grady's ten and six dimensional exceptional examples of irreducible holomorphic symplectic manifolds. Most of this note is a survey of work on these degenerations by Donagi-Ein-Lazarsfeld, de Cataldo-Maulik-Shen and Felissetti-Mauri, although in certain cases we provide details that were missing the previous articles.
In this paper we prove that the Gorenstein cyclic quotient singularities of type \frac 1l (1,..., 1,l-(r-1)) with $l\geq r\geq 2$, have a \textit{unique}torus-equivariant projective, crepant, partial resolution, which is ``full'' iff either $l\equiv 0$ mod $% (r-1) $ or $l\equiv 1$ mod $(r-1) $. As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of $\lfloor \frac{l}{r-1} \rfloor$ prime divisors, $\lfloor \frac{l}{r-1}\rfloor - 1$ of which are isomorphic to the total spaces of $\Bbb{P}_{\Bbb{C}}^1$-bundles over $\Bbb{P…
Monge-Ampère degenerations and conifold contractions in Fujiki class $\mathcal{C}$
针对具有普通二重点奇点的 Calabi-Yau 簇，在 Fujiki 类中建立例外轨迹外的度量一致估计，并证明三维情形下度量在 Gromov-Hausdorff 意义下收敛，推广了 Song 的工作。
Wahl singularities in degenerations of del Pezzo surfaces
本文通过引入带标记的 del Pezzo Wahl 链，刻画了次数为 $1 \leq \ell \leq 9$ 的 del Pezzo 曲面退化中的所有 Wahl 奇点，并揭示了其与伪加权射影平面的对应关系及在低次曲面中的应用。
Full exceptional collections for anticanonical log del Pezzo surfaces
本文受同调镜像对称启发，利用 McKay 对应与极小解消，为 Johnson–Kollár 对数 del Pezzo 曲面对应典范栈构造显式完全例外集，并刻画相关二次 del Pezzo 曲面。
We show that the K-moduli spaces of log Fano pairs $(\mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$ varies in the interval $(0,1)$. We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-O'Grady's prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $\ma…
Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem
本文研究了由Persson引入的一类canonical polarized表面，这些表面作为Galois覆盖P²的八条一般线的分支。通过构造紧化的模空间，并利用KSBA稳定对描述稳定退化，证明了紧化的模堆光滑。此外，建立了通用的全球Torelli型结果：通过反不变部分的第二上同调群的Hodge结构和相关的G作用，可以恢复大部分Persson表面。
Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations
本文提出不依赖 Torelli 定理的复 K3 曲面 Hodge 猜想构造性证明，借助至多十个 A_1-节点的四次退化、Clemens–Schmid 序列与 Picard–Lefschetz 理论，将任意有理 (1,1)-类实现为代数闭链。
Given a noncommutative partial resolution $A=\mathrm{End}_R(R\oplus M)$ of a Gorenstein singularity $R$, we show that the relative singularity category $Δ_R(A)$ of Kalck-Yang is controlled by a certain connective dga $A/^{\mathbb{L}}\kern -2pt AeA$, the derived quotient of Braun-Chuang-Lazarev. We think of $A/^{\mathbb{L}}\kern -2pt AeA$ as a kind of `derived exceptional locus' of the partial resolution $A$, as we show that it can be thought of as the universal dga fitting into a suitable recollement. This theoretical result has geometric consequences. When $R$ is an isolated hypersurface sing…
As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold $\mathbb{C}^3/\mathbb{Z}_4$, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface $\mathbb F_2$. We determine the explicit Kähler geometry of the smooth resolved manifold $Y$, which is the total space of the canonical bundle of $\mathbb F_2$. We study in detail the chamber structure of the space of st…
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