Sergey Gorchinskiy, Filippo Viviani
The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the (non-)existence of a tautological family over the coarse moduli space.
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We compute the Picard group of the moduli stack of stable hyperelliptic curves of any genus, exhibiting explicit and geometrically meaningful generators and relations.
Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.
We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers.
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
We construct a new compactification of the moduli space H_g of smooth hyperelliptic curves of genus g. We compare our compactification with other well-known remarkable compactifications of H_g .
Stable cohomology of moduli spaces of hyperelliptic curves on Hirzebruch surfaces
本文计算了 Hirzebruch 曲面上固定亏格超椭圆曲线模空间的稳定上同调，并用带标记点的非嵌入模空间进行描述。
By the geometry of the 3-fold quadric we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every $g \geq 2$.
Using the geometry of an almost del Pezzo threefold, we show that the moduli space of genus $g$ one-pointed ineffective spin hyperelliptic curves is rational for every $g\geq 2$.
Let $ε>0$. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve $C$ of genus $g$ over a finite field $k$ of cardinality $q$ given by $y^2+h(x)y=f(x)$ such that the $x$-coordinate map is ramified at $\infty$. In time $O(g^{2+ε} q^{1/2+ε})$ the algorithm outputs a set of generators of the Picard group $\mathrm{Pic}^0_k(C)$. This extends results which others have obtained when $g=1$. In this article we introduce a combinatorial tool, the `shape parameter', which we use together with character sum estimates from class field theory t…
There is a canonical isomorphism between the coarse moduli spaces of somooth hyperelliptic curves of genus g and binary forms of degree 2g+2 with nonzero discriminant. In this paper, we study the extension of this isomorphism to the compactification of these moduli spaces. The main result is that the canonical isomorphism above extends to a holomorphic map f_g from the compactication of the moduli space of hyperelliptic curves of genus g to the compactification of the moduli space of binary forms of degree 2g+2. Moreover, we work out how the boundary components are contracted by the map f_g.
The generating function for $S_n$-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g>1 is calculated. This answer generalizes the known ones for genera 2 and 3 and answers obtained by J. Bergstrom for any genus and n<8 points.
The Chow ring of the universal Picard stack over the hyperelliptic locus
本文确定了超椭圆轨迹上万有 Picard 层的有理 Chow 环及其生成元与关系，并计算了整 Picard 群，证明了其 Brauer 类阶由 $d-g$ 奇偶性决定。
The Desale-Ramanan Theorem is an isomorphism between the moduli space of rank two vector bundles over complex hyperelliptic curve and the variety of linear subspaces in an intersection of two quadrics. We prove a real version of this theorem for the moduli space of real vector bundles over a real hyperelliptic curve. We then apply this result to study the topology of the moduli space, proving that it is relatively spin and identifying the diffeomorphism type for genus two curves. Our results lay the groundwork for future study of the quantum homology of these moduli spaces.
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.
We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two parameters describing allowable singularities. By varying these parameters, we interpolate between the stacks of stable limits of A and D singularities and the quotients of the versal deformation spaces of A and D singularities by a natural $\mathbb{G}_m$ action.
This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of $\Mbar_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack $\Htilde_g^r$ of hyperelliptic $A_r$-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic $A_r$-stable curves. In particular, we prove that $\Htilde_g^r$ is a smooth algebraic stacks which can be described using cyclic covers of twisted curves of genus $0$ and it embeds in $\Mtilde_g^r$ (the moduli stack of $A_r$-stable curves) as the closure of the mod…
For a partition $lambda=\{lambda_1 \geq λ_2 \geq λ_3 \}$ of non-negative integers, we calculate the Euler characteristic of the local system $V_λ$ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some $λ$ of low degree, we make a guess for the motivic Euler characteristic of $V_λ$ using counting of curves over finite fields.
We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, parametrizing $n$-pointed irreducible nodal hyperelliptic curves, is generated by tautological divisors. Along the way, we improve Casnati's result that $\mathcal{H}_{g,n}$ is rational for $n \leq 2g+8$ to show $\mathcal{H}_{g,n}$ is rational for $n \leq 3g+5$.
The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two.
We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height $\mathcal h =1$.
In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.
Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure.
Let $\mathcal H_g$ be the moduli space of genus $g$ hyperelliptic curves. In this note, we study the locus $\mathcal L$ in $\mathcal H_g$ of curves admitting a $G$-action of given ramification type $σ$ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in $\mathcal L$. The proof of the results is based solely on representations of finite subgroups of $PGL_2 (\mathbb C)$ and the Riemann-Hurwitz formula.
In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $§$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $§$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in §$…
We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.
We speculate about an algebro-geometric proof of Harer's theorem on the rational Picard group of the moduli space of smooth complex curves. In particular, we refine the approach of Diaz and Edidin involving the Hurwitz space which parameterizes smooth covers of the projective line.
We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…
It is known that the moduli space $\overline{\mathcal{H}}_{g,n}$ of genus $g$ stable hyperelliptic curves with $n$ marked points is uniruled for $n \leq 4g+5$. In this paper we consider the complementary case. We calculate the canonical divisor of $\overline{\mathcal{H}}_{g,n}$ and show that it is effective for $n=4g+6$ and big for $n\leq 4g+7$. This leads us to conjecture that $\overline{\mathcal{H}}_{g,n}$ has non-negative Kodaira dimension for $n = 4g+6$ and is of general type for $n \geq 4g+7$.
We study the Picard groups of moduli spaces in positive characteristics and we give a "$p$-adic" proof that the Picard group of moduli of vector bundles of fixed determinant is isomorphic to the group of integers. Along the way we prove that the local fundamental group scheme of a normal unirational projective variety is trivial. This is reminiscent of results of Serre and Nygaard who studied the fundamental groups of smooth, projective, unirational varieties.
Let $X$ be a compact Riemann surface of genus $g\geq 3$ and let $\mathcal{M}_{\mathrm{par}}$ be the moduli space of semistable parabolic bundles over $X$. Let $\mathcal{M}_{\mathrm{parH}}$ denote the moduli space of semistable parabolic Higgs bundles over $X$. In this article, we study the Picard group of $\mathcal{M}_{\mathrm{par}}$ and $\mathcal{M}_{\mathrm{parH}}$.
We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the reduced automorphism group of $\p$ has at least two involutions, there exists a representative $C$ of the isomorphism class $\p$ which is defined over $L$.
We refine the notion of higher spin curves defined in terms of line bundles by Caporaso, Cornalba and Casagrande by adding the data of coherent nets of roots introduced by Jarvis in terms of torsion-free sheaves and we describe the boundary part of their moduli space. We provide then a presentation of the rational Picard group of this moduli space.
We determine the Brauer group and Picard group of the moduli space $U^{' s}_{L,par}$ of stable parabolic vector bundles of rank $r$ with determinant $L$ on a real curve $Y$ of arithmetic genus $g \,\ge\, 2$ with at most nodes as singularities.
We compute the intersection Betti numbers of the GIT model of the moduli space of Brill-Noether-Petri general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett-Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities.
We study the moduli space of smooth complete intersections of two quadrics in $\mathbb{P}^n$ by relating it to the geometry of the singular members of the corresponding pencils. Giving an alternative presentation for the moduli space of complete intersections, we compute the Picard group for all $n\geq 3$.
In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of n-pointed stable curves of genus g, for g greater than 2. These stacks are smooth, irreducible and have dimension 4g-3+n, yielding a geometrically meaningful compactification of the degree d universal Picard stack over the moduli stack of smooth curves with marked points.
We compute the Picard group of the moduli stack of smooth curves of genus $g$ for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on $\mathcal{M}_g$.
We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results ar…
We discuss the Picard group of moduli space $\mathcal{K}_g$ of quasi-polarized K3 surfaces of genus $g\leq 12$ and $g\neq 11$. In this range, $\mathcal{K}_g$ is unirational and a general element in $\mathcal{K}_g$ is a complete intersection with respect to a vector bundle on a homogenous space, by the work of Mukai. In this paper, we find generators of the Picard group $Pic_\mathbb{Q}(\mathcal{K}_g)$ using Noether-Lefschetz theory. This verifies the Noether-Lefschetz conjecture on moduli of K3 surfaces in these cases.
In this note, we prove that the Q-Picard group of the moduli space of n-pointed stable curves of genus g over an algebraically closed field is generated by the tautological classes. We also prove that the cycle map to the 2nd etale cohomology group is bijective.
In this paper, we study the Picard group of the moduli space of semistable sheaves on a smooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class. We focus especially on moduli spaces of sheaves of small discriminant, where we observe new and interesting behavior. Our method relies on constructing certain resolutions for semistable sheaves and applying techniques of geometric invariant theory to the resulting families of sheaves.
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…
In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-…
We compute the Picard group of the moduli space $U'$ of semistable vector bundles of rank $n$ and degree $d$ on an irreducible nodal curve $Y$ and show that $U'$ is locally factorial. We determine the canonical line bundles of $U'$ and $U'_L$, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification of $U'$.
We construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of Poincaré bundles over the…
Let $X$ be an irreducible smooth projective curve of genus $g \geq 2$ over $\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let $\mathrm{M}_{G, {\rm Higgs}}^δ$ be the moduli space of semistable principal $G$--Higgs bundles on $X$ of topological type $δ\in π_1(G)$. In this article, we compute the fundamental group and Picard group of $\mathrm{M}_{G, {\rm Higgs}}^δ$.
Integral Picard group of moduli of polarized K3 surfaces
本文研究固定次数且至多含有理二重点的极化 K3 曲面模叠，确定其整 Picard 群，并推广到准极化 K3 曲面叠与叠式周期区域。
Let F be an algebraically closed field with char(F) not equal to 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ιbe the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/<ι> is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/<ι> is cyclic.
On the Picard numbers of moduli spaces of one-dimensional sheaves on surfaces
本文研究曲面上支撑在充分正除子类中的一维稳定层模空间的皮卡数，给出一般渐近下界，并在特殊情形下借助稳定对模空间与相对希尔伯特概形证明该下界可达，同时举例说明渐近不可约性可能失效。
On K-moduli of Fano threefolds with degree 28 and Picard rank 4
本文研究次数28、Picard秩4的Fano三维簇的K-模，通过构造奇异环面K-多稳定Fano三维簇的代数形变，证明对应光滑Fano三维簇的K-模不可约分支为有理曲面。
This article treats the Picard group of the moduli (stack) of r-spin curves and its compactification. Generalized spin curves, or r-spin curves are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because they are the subject of a remarkable conjecture of E. Witten, and because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. We generalize results of Cornalba, giving relations between many of the elements of the Picard group of the stacks. These relations a…
Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves
该研究针对曲面上的希格斯层模空间与希钦映射，证明了高维谱簇的Noether–Lefschetz型定理，给出一般希钦纤维非空的充要条件，并分析第二陈类变化对模空间几何的影响。
On the Picard Group of the Moduli Space of Curves via $r$-Spin Structures
本研究通过推导曲线模空间二阶有理上同调中Pandharipande-Pixton-Zvonkine关系的显式表达式，结合Arbarello-Cornalba定理证明了对应情形下的Pixton猜想。
We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole $Sp(20,R)$. We construct an interesting submoduli, which we call \emph{hyperelliptic locus}, over which the weight 3 $Q$-Hodge structure is the third wedge product of the weight 1 $Q$-Hodge structure on the corresponding hyperelliptic curve. The non-extendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved.
Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L…
Hyperelliptic curves on $K3$ surfaces
该研究证明皮卡秩为1的K3曲面上任意多个一般点可由超椭圆曲线连接，得到小对角的Beauville–Voisin分解及挠阶新界，还将Corvaja–Zucconi定理推广到任意次数K3曲面，证明其二次点在有限扩张后稠密。
Stratified Hyperbolicity of the moduli stack of stable minimal models, II: Big Picard Theorem and the stratified Brody hyperbolicity
本文研究稳定极小模型模叠的层状双曲性，通过引入双有理容许分层，证明每层满足大皮卡定理，进而得出每层是Borel双曲和Brody双曲的。
We prove that, for a natural class of Bridgeland stability conditions on $\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after "rotation") equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projectiv…
The Kodaira classification of the moduli of hyperelliptic curves
本文研究带标记点的超椭圆曲线模空间的双有理几何，完成了其Kodaira分类，并分析了奇点性质及伪有效锥的结构。
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