Michela Artebani
S. Kondō defined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of $\mathcal M_3$ and describe its geometry.
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In this paper, we study all ways of constructing modular compactifications of the moduli space $\mathcal{M}_{g,n}$ of $n$-pointed smooth algebraic curves of genus $g$ by allowing markings to collide. We find that for any such compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications of $\mathcal{M}_{g,n}$ with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifica…
Hurwitz correspondences are certain multivalued self-maps of the moduli space $\mathcal{M}_{0,N}$. They arise in the study of Thurston's topological characterization of rational functions. We consider the dynamics of Hurwitz correspondences and ask: On which compactifications of $\mathcal{M}_{0,N}$ should they be studied? We compare a Hurwitz correspondence $\mathcal{H}$ across various modular compactifications of $\mathcal{M}_{0,N}$, and find a weighted stable curves compactification $X_N^\dagger$ that is optimal for its dynamics. We use $X_N^\dagger$ to show that the $k$th dynamical degree o…
We show that a certain moduli space of minimal $A_\infty$-structures coincides with the modular compactification $\bar{\mathcal{M}}_{1,n}(n-1)$ of $\mathcal{M}_{1,n}$ constructed by Smyth. In addition, we describe these moduli spaces and the universal curves over them by explicit equations, prove that they are normal and Gorenstein, show that their Picard groups have no torsion and that they have rational singularities if and only if $n\le 11$.
We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 4. We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.
Let $G$ be a connected reductive algebraic group. Let $\mathcal{E}\rightarrow \mathcal{B}$ be a principal $G\times G$-bundle and $X$ be a regular compactification of $G$. We describe the Grothendieck ring of the associated fibre bundle $\mathcal{E}(X):=\mathcal{E}\times_{G\times G} X$, as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle on $\mathcal{B}$. These are relative versions of the results on equivariant $K$-theory of regular compactifications of $G$. They also generalize the well known results on the Grothendieck rings of projective bundles, toric bu…
We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to …
Integral Chow rings of modular compactifications of $\mathcal{M}_{1,n\leq 6}$
本文计算了n≤6时所有仅参数化Gorenstein曲线及其光滑不同标记的模尔紧化的积分柴环，包括Deligne-Mumford、Schubert和Smyth紧化等，并研究了柴- Künneth生成性质及Getzler关系。
Relative compactifications of semiabelian Néron models, I
本文研究完全退化半阿贝尔Néron模型的相对紧化问题，证明了满足特定Cohen-Macaulay性质与丰富可逆层条件的唯一相对紧化的存在性。
In this paper we compute the multiple cover Gromov-Witten integrals (analog of the Aspinwall-Morrison formula) for the unramified compactification of the moduli space of stable maps to an embedded $\PO$ in a Calabi-Yau threefold $X$ with the normal bundle $\mathcal{O}_{\PO}(-1) \bigoplus \mathcal{O}_{\PO}(-1)$.
We define the wonderful compactification of an arrangement of subvarieties. Given a complex nonsingular algebraic variety $Y$ and certain collection $\mathcal{G}$ of subvarieties of $Y$, the wonderful compactification $Y_\mathcal{G}$ can be constructed by a sequence of blow-ups of $Y$ along the subvarieties of the arrangement. This generalizes the Fulton-MacPherson configuration spaces and the wonderful models given by De Concini and Procesi. We give a condition on the order of blow-ups in the construction of $Y_\mathcal{G}$ such that each blow-up is along a nonsingular center.
For a stable curve of genus $g\geq 2$ and simple Lie algebra of type A or C, we show that the conformal blocks algebra $\mathcal{A}$ on $\overline{\mathcal{M}}_g$ is finitely generated and establish an explicit connection to Schmitt and Muñoz-Castañeda's compactification of the moduli space of $G$-bundles.
Relative compactification of semiabelian Néron models, II
本文研究半阿贝尔Néron模型的相对紧化问题，证明了满足特定Cohen-Macaulay性与层条件的唯一相对紧化的存在性，先后讨论部分退化情形及基环为Dedekind整环的情形，补充了此前完全退化情形的相关研究。
We study the geometry of Gorenstein curve singularities of genus two, and of their stable limits. These singularities come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1\leq m <n$, a stability condition - using one of the markings as a reference point, and therefore not $\mathfrak S_n$-symmetric - defines proper Deligne-Mumford stacks $\overline{\mathcal M}_{2,n}^{(m)}$ containing the locus of smooth curves as a dense open substack.
We show that Martin Olsson's compactification of moduli space of polarized abelian varieties in \cite{ols08} can be interpreted in terms of KSBA stable pairs. We find that there is a canonical set of divisors $S(K_2)$ associated with each cusp. Near the cusp, a polarized semiabelic scheme $(\mathcal{X}, G,\mathcal{L})$ is the canonical degeneration given by the compactification if and only if $(\mathcal{X},G,Θ)$ is an object in $\overline{\mathscr{AP}}_{g,d}$ for any $Θ\in S(K_2)$. Moreover, we give an alternative construction of the compactification by using mirror symmetry. We construct a to…
Let $(X,H)$ be a smooth, projective, polarized surface over $\mathbb{C}$, and let $v \in K_{\mathrm{num}}(X)$ be a class of positive rank. We prove that for certain Bridgeland stability conditions $σ= (\mathcal{A}, Z)$ "on the vertical wall" for $v$, the good moduli space $M^σ(v)$ parameterizing S-equivalence classes of $σ$-semistable objects of class $v$ in $\mathcal{A}$ is projective. Moreover, we construct a bijective morphism $M^{\mathrm{Uhl}}(v) \to M^σ(v)$ from the Uhlenbeck compactification of $μ$-stable vector bundles.
We study the Chow groups and the Chow motives of the wonderful compactifications $Y_{\mathcal{G}}$ of arrangements of subvarieties. We prove a natural decomposition of the Chow motive of $Y_\mathcal{G}$, in particular of the Fulton-MacPherson configuration space $X[n]$. As a consequence, we prove a decomposition of the Chow motive of $X[n]/\frak{S}_n$. A generating function for the Chow groups and for the Chow motive of $X[n]$ is given.
We describe compactifications of moduli spaces of Burniat surfaces with $2\leq K_{X}^{2}\leq5$ obtained by adding KSBA surfaces, i.e. slc surfaces $X$ with ample canonical class $K_{X}$.
The goal of the present paper is to construct a smooth compactification of the moduli superstack classifying pointed $\mathcal{N} =1$ SUSY (= $\text{SUSY}_1$) curves. This construction is based on the Abramovich-Jarvis-Chiodo compactification of the moduli stack classifying spin curves. First, we give a general framework of a theory of log superschemes (or more generally, log superstacks). Then, we introduce the notion of a pointed (stable) log twisted $\text{SUSY}_1$ curve; it may be thought of as a logarithmic and twisted generalization of the classical notion of a pointed $\text{SUSY}_1$ cu…
The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of th…
Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification the moduli space of principally polarized abelian varieties of dimension g stabilizes as g grows and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebro-geometric proof of their theorem that also takes into account the mixed Hodge structure that is present here. We find the latter to be impure.
We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $\overline{\mathcal A}_3$ of the moduli space ${\mathcal A}_3$ of complex principally polarized abelian threefolds, and we conjecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $g\ge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone toroidal compactification of ${\mathcal A}_g$ for any $g\ge 3$.
We describe a compactification by stable pairs (also known as KSBA compactification) of the $4$-dimensional family of Enriques surfaces which arise as the $\mathbb{Z}_2^2$-covers of the blow up of $\mathbb{P}^2$ at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has …
Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians
本文研究了对称矩阵空间的紧化与指向Kontsevich空间的等价格拉斯曼流形，推导了关于锥形空间的理性几何结论。
We classify the Deligne-Mumford stacks M compactifying the moduli space of smooth $n$-pointed curves of genus one under the condition that the points of M represent Gorenstein curves with distinct markings. This classification uncovers new moduli spaces $\bar{\mathcal{M}}_{1,n}(Q)$, which we may think of coming from an enrichment of the notion of level used to define Smyth's $m$-stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the $\bar{\mathcal{M}}_{1,n}(Q)$'s, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model …
We study the cones in the first Voronoi or perfect cone decomposition of quadratic forms with respect to the question which of these cones are basic or simplicial. As a consequence we deduce that the singular locus of the moduli stack ${\mathcal A_g^{\mathop{Perf}}}$, the toroidal compactification of the moduli space of principally polarized abelian varieties of dimension $g$ given by this decomposition, has codimension $10$ if $g \geq 4$. Moreover we describe the non-simplicial locus in codimension $10$. We also show that the second Voronoi compactification ${\mathcal A_{g}^{\mathop{Vor}}}$ h…
We study the moduli space $\mathcal{F}_{T_1}$ of quasi-trielliptic K3 surfaces of type I, whose general member is a smooth bidegree $(2,3)$-hypersurface of $\mathbb{P}^1\times \mathbb{P}^2$. Such moduli space plays an important role in the study of the Hassett-Keel-Looijenga program of the moduli space of degree $8$ quasi-polarized K3 surfaces. In this paper, we consider several natural compactifications of $\mathcal{F}_{T_1}$, such as the GIT compactification and arithmetic compactifications. We give a complete analysis of GIT stability of $(2,3)$-hypersurfaces and provide a concrete descript…
We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,εR)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.
It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.
Explicit KSBA compactifications of moduli spaces of secondary and tertiary Burniat surfaces
本文针对次数 $K^2=5$、$4$、$3$ 的次级与三级 Burniat 曲面模空间，给出通过添加典范类丰富的 slc 曲面所得的显式几何 KSBA 紧化。
Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. We are interested in the branch locus ${\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of $\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in $\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for $g\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\bar{\mathcal{M}_{g}}$. To do so we exhi…
The new compactification of moduli scheme of Gieseker-stable vector bundles with the given Hilbert polynomial on a smooth projective polarized surface (S;H), over the field k = \bar k of zero characteristic, is constructed in previous papers of the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the schemes which are certain modifications of S. We obtain the class of modified surfaces to appear in the construction. Keywords: moduli space, semistable coherent sheaves, blowup algebra, algebraic surface.
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…
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space, one for the ramification divisor and another for the rational curve divisor. In the course of the proof, we further develop the theory of integral affine spheres with 24 singularities. We also construct moduli of rational (generalized) elliptic stable slc surfaces of types ${\bf A_n}$ ($n\ge1$), ${\bf C_n}$ ($n\ge0$) and …
We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb{P}^1\times\mathbb{P}^1$ branched along a specific $(4,4)$ curve. We show that, up to a finite group action, this stable pair compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb{P}^1)^8//\mathrm{SL}_2$ with the symmetric linearization.
The automorphism groups of generalized Kausz compactifications and spaces of complete collineations
本文研究了广义Kausz紧化与完整共线空间的自同构群，通过反canonical包的正性确定了其结构，部分结果基于先前的工作。
A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces.
Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces
本文针对 Campedelli 曲面与 Burniat 曲面模空间的两个连通分支，通过加入典范类丰富的 slc 曲面，给出了其几何紧化的显式构造。
Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism
该研究针对具有3阶和4阶非辛自同构（不动点含亏格≥2曲线）的K3曲面模空间，描述其Baily-Borel、环面及KSBA几何紧化，对3阶处理所有最大维族，证明环面与KSBA紧化可通过ADE根格简单描述。
In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example the compactification of the space of stable rank-2 vector bundles with Chern classes $c_1=0, c_1=2$ on the projective plane is studied in more detail. Proofs are only indicated and wi…
Gorenstein-Fano polytopes and compactifications of rank 2 polyptych lattices
本文构造并计算具有两个图卡的秩 2 polyptych 格族，展示其非经典凸几何、相关 Gorenstein-Fano 多面体及去热带化紧化的 Cox 环表示。
Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs (X,H) consisting of a degree two K3 surface X and an ample divisor H. Specifically, we construct and describe explicitly a geometric compactification $\bar{P_2}$ for the moduli of degree two K3 pairs. This compactification has a natural forgetful map to the Baily-Borel compactification of the moduli space …
Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability
本文利用K-稳定性和线构形研究光滑del Pezzo曲面模空间的紧化，构造了log del Pezzo对的K-模空间，并证明了其在$d=2,3,4$时同构于del Pezzo曲面的K-模空间，而在$d=1$时两者不同。
Extending the Torelli map to alternative compactifications of the moduli space of curves
本文研究了具有轴状奇点的曲线，构造了替代紧化并证明Torelli映射可扩展，同时识别了Smyth意义下的替代紧化中轴状区域。
We construct various modular compactifications of the space of elliptic K3 surfaces using tools from the minimal model program, and explicitly describe the surfaces parametrized by their boundaries. The coarse spaces of our constructed compactifications admit morphisms to the Satake-Baily-Borel compactification. Finally, we show that one of our spaces is smooth with coarse space the GIT quotient of pairs of Weierstrass K3 surfaces with a chosen fiber.
When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic
We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers i…
Moduli of Nodal Sextic Curves via Periods of K3 Surfaces
本文研究了节点六次曲线的模空间，通过将其实现为IV型算术商的开放子空间，证明了GIT紧化与Looijenga紧化之间的同构，并将结果推广至具有特定对称性的情形。
We classify all normal G^2_a-surfaces with Picard number one, and characterize which of these surfaces have at worst log canonical, and which have at worst log terminal singularities, answering a question of Hassett and Tschinkel (Int. Math. Res. Not., 1999). We also find all G^2_a-structures on these surfaces and show that these surfaces and their minimal desingularizations have the same G^2_a-structures (modulo equivalence of G^2_a-actions). In particular, we show that some of these surfaces admit one dimensional moduli of G^2_a-structures, answering another question of Hassett and Tschinkel…
We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe…
We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.
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.
Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-Borel compactification of the moduli space polarized K3's of degree 2, and also of the period map of cubic fourfolds. On the other hand, the period map of the GIT moduli space of quartic surfaces is significantly more subtle. In our previous paper, we introduced a Hassett-Keel-Looijenga program for certain locally symmetric varieties of Type IV. As a consequence, we gave a complete conjectural decomposition in…
Cohomology of symplectic $T^n$ - reductions and compactifications of $\mathcal{M}_{0, n}$
本文通过室分解和Kirwan-Goldin结果，研究了复格拉斯曼流形上辛约简的上同调环，并探讨了$\mathcal{M}_{0,n}$不同紧化的上同调结构。
We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor, and allow both the surface and the curve to degenerate. For plane curves of degree d at least 4, we obtain a compactification M_d which is a moduli space of stable pairs (X,D) using the log minimal model program. A stable pair (X,D) consists of a surface X such that -K_X is ample and a divisor D in a given linear system on X with specified singularities. No…
Principal bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties
本文研究了在积分典范模型的托里奥利紧致化上，主$\mathcal{G}^c$-和$M^c$-丛的典范扩展构造方法。
We construct several examples of genus-one fibered K3 surfaces without a global section with type $I_{n}$ fibers, by considering double covers of a special class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2. The resulting K3 surfaces have bisection geometries. F-theory compactifications on these K3 genus-one fibrations without a section times a K3 yield models that have $SU(n)$ gauge symmetries with a discrete $\mathbb{Z}_2$ symmetry.
Non-isomorphic smooth compactifications of the moduli space of cubic surfaces
针对复三次曲面模空间的Kirwan爆破与环面紧化因上同调相同是否同构的问题，证明二者在Grothendieck环中等价但非同构、非$K$-等价，并建立相关奇点与典范类的研究技术。
Metric Degenerations and Satake Compactification of Enriques Surfaces
本研究确定Enriques曲面上单位直径Ricci平坦Kähler度量模空间的Gromov–Hausdorff紧化，证明其度量退化由伴随Satake紧化编码，分类了所有相关GH极限。
We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric invariant theory. We illustrate this with the moduli spaces of smooth quartic curves and rational elliptic surfaces.
The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic stable bundles of rank 2 and also prove the existence of components with smooth points.
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