Sergey Gorchinskiy, Filippo Viviani
We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last case we use the description of the relative Picard group of the universal family of trigonal curves.
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Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4. We show that the canonical model of $X$ is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of $X$. This surface allows us to define four invariants of $X$ and hence to stratify the moduli space of four-gonal curves by means of closed irreducible subvarieties whose dimensions we compute.
The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $ν$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which…
Harris and Morrison constructed semistable families f:F \to Y of k-gonal curves of genus g such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus in the moduli space. Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio K_F^2 / χ(O_F) is 8 asymptotically with respect to g(X). We show also that if the gonality is maximal and some conjectured estimates of Harris and Morrison hold, the slope of the fibration f: F\to Y is 12 asymptotically with respect to g and…
Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ is of type $(0;c_1,...,c_k)$, where some $c_i$ appears an odd number of times. This result is applied to $q$-gonal curves and to plane quartics. For $q$-gonal curves, we prove that non-normal $q$-gonal curves can be defined over their field of moduli and we construct examples of normal $q$-gonal curves with field of moduli $\mathbb{R}$ that can not be defined over $\ma…
There is no stable Siegel modular form that vanishes on the trigonal locus in every moduli space of curves.
The $d$-gonal locus in the moduli space of tropical plane curves
本文研究了热带平面曲线的gonality与牛顿多边形预期gonality的关系，证明了在高 genus 下两者维数一致。
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…
We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we provide.
Let $M(n,ξ)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $ξ$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke morphisms to describe a smooth open set and to compute the dimension of a component of the Hilbert scheme $Hilb_{M(n,ξ)}$, of the scheme of morphisms $Mor(\mathbb{G},M(n,ξ))$ and of the moduli space $ M_{X \times \mathbb{G}}$ of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove th…
We study noncompact Calabi-Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi-Yau threefolds that have infinitely many distinct deformations, constructing them explicitily, and describe the effect that such deformations produce on moduli spaces of vector bundles.
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.
It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has been put forward by Mercat, who conjectured that the second Clifford index of a curve should be equal to its classical Clifford index, defined in terms of gonality. Using moduli of sheaves on generic K3 surfaces, we prove Mercat's conjecture for generic curves of every genus. Furthermore, for odd g we identify an effective divisor in the moduli space M_g along …
The line bundles on the moduli stack of principal bundles on families of curves
研究连通约化群G在任意光滑曲线族上主丛模叠的Picard群，确定其线丛结构。
This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then the families are equivalent to morphisms of curves into M. In order to have complete moduli for such families, one needs to compactify the stack of stable maps into M. It turns out that in the boundary, the curve must acquire extra structure and you'd better read the paper to see what that structure is. Applications to fibered surfaces (see math.AG/9804097), a…
We study branched covers of curves with specified ramification points, under a notion of equivalence derived from linear series. In characteristic 0, no non-constant families of covers with fixed ramification points exist. In positive characteristic we formulate a necessary and sufficient condition for the existence of such a family. We unconditionally prove one direction of this conjecture, and by studying infinitesimal deformations show the other direction in characteristic 2 and 3.
MMP for locally stable families and wall crossing for moduli of stable pairs
本文研究稳定对模空间随系数变化的约化与穿墙态射，通过在正规局部稳定族上运行带缩放的MMP，将已有klt情形的结果推广到更一般的局部稳定情形。
Let V^{r}_{d,g, δ} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with δnodes. Under suitable numerical assumptions on d and g, for every 0 \le δ\le g we construct an irreducible component of V^{r}_{d,g, δ} having the expected number of moduli.
Let $H$ be a connected semisimple linear algebraic group defined over $\mathbb C$ and $X$ a compact connected Riemann surface of genus at least three. Let ${\mathcal M}'_X(H)$ be the moduli space parametrising all topologically trivial stable principal $H$-bundles over $X$ whose automorphism group coincides with the centre of $H$. It is a Zariski open dense subset of the moduli space of stable principal $H$-bundles. We prove that there is a universal principal $H$-bundle over $X\times {\mathcal M}'_X(H)$ if and only if $H$ is an adjoint group (that is, the centre of $H$ is trivial).
Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus g=(n-1)(n-2)/2-d-k. We define the "number of moduli of W" as the dimension of the image of W with respect to this map. If W has the expected dimension equal to 3n+g-1-k, then the number of moduli of W is at most equal to the min(3g-3, 3g-3+ρ-k), dove ρis the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that W…
Iterated Grassmannian bundles over moduli stacks of vector bundles on a curve are shown to be birational to an affine space times a moduli stack of degree 0 vector bundles, following the method of King and Schofield. Applications include the birational type of some Brill-Noether loci, of moduli schemes for vector bundles with parabolic structure or with level structure and for A. Schmitt's decorated vector bundles. A further consequence concerns the existence of Poincare families on finite coverings of the moduli schemes.
We establish a new moduli theory for divisors, that interpolates between the Hilbert scheme and the Cayley-Chow variety. This completes the last step in the construction of a good moduli theory for stable pairs $(X,Δ)$.
We describe a systematic way of constructing effective divisors on the moduli space of stable curves of genus g having exceptionally small slope. We prove that any divisor on \bar{M}_g consisting of curves failing a certain Green-Lazarsfeld syzygy type condition, provides a counterexample to the Harris-Morrison Slope Conjecture. These divisors generalize our original isolated counterexample to the Slope Conjecture which was the divisor on M_{10} of curves lying on K3 surfaces. We also introduce a new stratification of M_g, somewhat similar to the classical stratification given by gonality, but…
Geometric Shafarevich boundedness conjecture for families of polarized varieties
本文研究极小稳定模型模栈上的几何沙法列维奇有界猜想，主要贡献是证明了该猜想的成立。
Wall-crossing for K-moduli spaces of certain families of weighted projective hypersurfaces
本文研究了特定加权投影超曲面的K-模空间，并通过显式分析展示了K-模空间的墙交叉现象，揭示了其与GIT变化的关系及最后的分式收缩。
The main subject is the difference between the coarse moduli space and the stack of hyperelliptic curves. In particular, we compute their Picard groups, giving explicit description of the generators. We also study how many families of hyperelliptic curves may have the same modular map as well as the existence of the tautological family over the coarse moduli space of hyperelliptic curves.
Tropical Donagi theorem
本文针对热带四角构造，证明其为保持普里姆簇的三元组变换，证实前期推测并得到热带普里姆-托雷利态射非单射的新结果，同时指出该构造在边收缩下的缺陷。
This paper deals with a study of the rational elliptic surfaces whose $J$-invariant functions are of degree one. Almost all of these elliptic surfaces have four singular fibers, while the remaining surfaces have only three singular fibers. The moduli space of these elliptic surfaces is canonically isomorphic to the projective line by taking the $J$-values for a certain fixed type of singular fibers. Over the moduli space, we discuss our elliptic surfaces, and investigate how their sections are described by the parameter of the moduli space. By using a covering space of the moduli, we construct…
We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.
Moduli of curves and moduli of sheaves
本文综述了三维簇上曲线与层模空间间的GW/DT对应，涵盖后代与相对对应及其在Calabi-Yau五次三维簇上的证明，并展望了族中的猜想框架。
We prove in one go that each of the 4 families of Burniat surfaces with K^2 = 6,5,4, is a connected component of the moduli space of surfaces of general type. We prove also the rationality of each component. In the nodal case (one of the two families for K^2_S = 4) a very surprising and new phenomenon occurs. Both the moduli space for the minimal models and the Gieseker moduli space for canonical models are everywhere non reduced. But the nilpotence order is higher for the first.
Rational curves on a smooth Hermitian surface II: moduli and counting of certain infinite families
研究光滑\(k -\)埃尔米特曲面上由特定四项多项式参数化的曲线，确定其\(k -\)射影等价类、模空间及各轨道曲线数量，还给出曲线的光滑性、自反性和最小定义域等。
We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.
In this paper we study four families of moduli problems which give rise to two dimensional examples of the Hitchin map. Using a few Fourier-Mukai transforms on the corresponding spectral curves, we give isomorphisms between these moduli problems.
We briefly survey recent results related to linear series on curves that are general in various moduli spaces, highlighting the interplay between algebraic geometry on a general curve and the combinatorics of its degenerations. Breakthroughs include the proof of the Maximal Rank Theorem, which determines the Hilbert function of the general linear series of given degree and rank on the general curve in M_g, and complete analogs of the standard Brill-Noether theorems for curves that are general in Hurwitz spaces. Other advances include partial results in a similar direction for linear series in …
Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds
本文证明Gushel-Mukai簇与其超平面截面的Kuznetsov分量间拉回与推出函子保持Bridgeland稳定性，并由此构造GM四维簇模空间的8维Lagrangian族及证明GM三维簇模空间的射影性。
In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type.
This paper provides explicit closed formulas in terms of tautological classes for the cycle classes of the height and Artin invariant strata in families of K3 surfaces. The proof is uniform for all strata and uses a flag space as the computations in [arXiv:math/0412272] for the Ekedahl-Oort strata for families of abelian varieties, but employs a Pieri formula formula to determine the push down to the base space.
On the distribution of non-rigid families in the moduli spaces
本文研究模空间中非刚性族的分布，基于Shimura品种的类比，证明在满足条件时非刚性类形映射像包含在Hodge locus中，并通过闭子方案得到几何Bombieri-Lang有限性定理。
Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U of S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily-Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.
We prove that a complete family of linearly non-degenerate rational curves of degree $e > 2$ in $\mathbb{P}^n$ has at most $n-1$ moduli. For $e = 2$ we prove that such a family has at most $n$ moduli. It is unknown whether or not this is the best possible result. The general method involves exhibiting a map from the base of a family $X$ to the Grassmaninian of $e$-planes in $\mathbb{P}^n$ and analyzing the resulting map on cohomology.
Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincaré bundles over open subsets of this moduli space, and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan and Balaji-Biswas-Nagaraj-Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics.
Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.
There are no exotic compact moduli of sheaves on a curve
研究曲线上给定次数和正秩的凝聚层模空间，证明仅恢复经典斜率半稳定向量丛模空间的非空开条件能产生泛闭充分模空间，解决层模空间的泛闭充分性问题。
We study the Severi variety $V_{d,g}$ of plane curves of degree $d$ and geometric genus $g$. Corresponding to every such variety, there is a one-parameter family of genus $g$ stable curves whose numerical invariants we compute. Building on the work of Caporaso and Harris, we derive a recursive formula for the degrees of the Hodge bundle on the families in question. For $d$ large enough, these families induce moving curves in $\bar{M}_g$. We use this to derive lower bounds for the slopes of effective divisors on $\bar{M}_g$. Another application of our results is to various enumerative problems …
Families of stable curves of genus $γ$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_γ$. It is natural to compactify the corresponding moduli problem using stable maps into the stack. In order to get a complete moduli problem, the source curves must acquire extra structure, and you will have to read the paper to find what that is. In this paper, we define these stable maps into $\bar{\cal M}_γ$ in characteristic 0, and show that they form a proper Deligne-Mumford stack admitting a projective coarse moduli space. A comparison with Al…
We study classes $P_{g,T}(α;β)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. A comparison with classes $Q_{g,T}$ arising from sections of the universal Jacobian shows the classes $P_{g,T}(α;β)$ are polynomial in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.
In this paper we proof the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over $\overline{M}_{g}$ to the construction of the universal moduli space of swamps.
We study the geometry of a class of $n$-dimensional smooth projective varieties constructed by Schreieder for their noteworthy Hodge-theoretic properties. In particular, we realize Schreieder's surfaces as elliptic modular surfaces and Schreieder's threefolds as one-dimensional families of Picard rank $19$ $K3$ surfaces.
We introduce a formalism of descent of moduli spaces, and use it to produce limit linear series moduli spaces for families of curves in which the components of fibers may have monodromy. We then construct a universal stack of limit linear series over the stack of semistable curves of compact type, and produce new results on existence of real curves with few real linear series.
Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If …
Connecting families of curves
本文研究光滑射影簇上连接多个一般点的曲线族，在 Kodaira 维数非负时给出亏格线性下界，并刻画连接族 gonality 的极限值及高次超曲面的渐近行为。
We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and Piyaratne-Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers. Our main application is the generalization of Mukai's theory for moduli spaces of semistable sheaves on…
Maximal Variation in the Moduli of Curves
本文研究了曲线模空间中最大变化区域的性质，揭示了非Gorenstein奇点对退化的影响，并建立了退化区域的分类方法。
Kuga-Satake construction on families of K3 surfaces of Picard rank 14
利用IV$_6$型与II$_4$型Hermite对称域间的等距，通过Kuga-Satake构造建立了Picard秩14的K3曲面模空间与具有全确定四元数乘法的极化阿贝尔8-fold模空间之间的几何映射，并展示了该映射在具体族上的实现。
We propose an alternative definition for families of stable pairs $(X,D)$ over a possibly non-reduced base when $D$ is reduced, by replacing $(X,D)$ with an appropriate orbifold pair $(\mathcal X,\mathcal D)$. This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Moreover, adjunction for $(\mathcal X,\mathcal D)$ holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial …
In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlevé equations. The classification of ten families is given by considering the Riemann-Hilbert morphism from a moduli space of connections with certain type of regular and irregular singularities to a corresponding catetorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), which is called the monodromy space. Explicit equations of the monodromy spaces for ten families are calculated. Moreover, we obtain natural explicit f…
Tautological families of cyclic covers of projective spaces
本文研究射影空间循环覆盖的tautological族存在性，给出存在条件并确定相关模空间的有理性，受超椭圆曲线模空间工作启发。
We give explicit descriptions of some Noether-Lefschetz divisors in the moduli space of Gushel-Mukai fourfolds. As a consequence we obtain that their Kodaira dimension is negative.
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex $V$-spaces (a generalization of complex $V$-manifolds in the sense of Satake). As an application, we prove the following hyperbolic version of Campana's isotriviality conjecture: for the smooth family of canonically polarized or polarized Calabi-Yau manifolds, when the Kobayashi pseudo-distance of the base vanishes identically, the family must be isotrivial, that is, any two fibers are isomorphic. We also prove that for the smooth pr…
In the infinite series of complete families of Calabi-Yau manifolds $\tilde{f}_n: \tilde{\mathcal{X}}_n\rightarrow \mathfrak{M}_{n, n+3}$, where $n$ is an odd number, arising from cyclic covers of $\mathbb{P}^n$ branching along hyperplane arrangements (\cite{SXZ13}), the set of CM points is dense for $n=1, 3$ and finite for $n\geq 5$.
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