Hausel introduced a commutative algebra -- the multiplicity algebra -- associated to a fixed point of the C^*-action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in the isomorphism class of the algebra.
Some divisor class relations are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target.
We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.
The aim of this note is to describe the restriction map from the moduli space of stable rank 2 bundle with small $c_2$ on a jacobian $X$ of dimension 2, to the moduli space of stable rank 2 bundles on the corresponding genus 2 curve $C$ embedded in $X$.
We study the Buchsbaum-Rim multiplicity, ab inicio, through a Koszul-Čech spectral sequence. We show that the Buchsbaum-Rim multiplicity is the arithmetic genus (Euler characteristic) of Koszul homology sheaves on a projective space over the base scheme.
We compute the stringy E function of the moduli space of rank 2 bundles over a Riemann surface of genus 3. In doing so, we answer a question of Batyrev about the stringy E functions of the GIT quotients of linear representations.
We study the stability of the normal bundle of canonical genus $8$ curves and prove that on a general curve the bundle is stable. The proof rests on Mukai's description of these curves as linear sections of a Grassmannian $\mathrm{G}(2,6)$. This is the next case of a conjecture by M. Aprodu, G. Farkas, and A. Ortega: the general canonical curve of every genus $g \geq 7$ should have stable normal bundle. We also give some more evidence for this conjecture in higher genus.
Let $π:X\rightarrow \mathbb{P}^2$ be a K3 surface of genus 2 and $L=π^{\ast}\mathcal{O}_{\mathbb{P}^2}(3)$, and assume that $π^{\ast}\mathcal{O}_{\mathbb{P}^2}(1)$ is ample as a line bundle on $X$. In this paper, we give a numerical characterization of initialized and ACM line bundles on $X$ with respect to $L$ and construct families of semistable indecomposable ACM bundles of higher rank, by using extensions of ACM line bundles.
In this note we describe a unique linear embedding of a prime Fano 4-fold F of genus 10 into the Grassmannian G(3,6). We use this to construct some moduli spaces of bundles on linear sections of F. In particular the moduli space of bundles with Mukai vector (3,L,3) on a generic polarized K3 surface (S,L) of genus 10 is constructed as a double cover of the projective plane branched over a smooth sextic.
Let X be a complex projective curve which is smooth and irreducible of genus 2. The moduli space M_2 of semistable symplectic vector bundles of rank 4 over X is a variety of dimension 10. After assembling some results on vector bundles of rank 2 and odd degree over X, we construct a generically finite cover of M_2 by a family of 5-dimensional projective spaces, and outline some applications.
Let U(r) be the moduli space of rank r vector bundles with trivial determinant on a smooth curve of genus 2. The map theta_r: U(r) -> |r Theta|, which associates to a general bundle its theta divisor, is generically finite. In this paper we give a geometric interpretation of the generic fibre of theta_r.
Building on our previous joint work with A. Schmitt [7] we explain a recursive algorithm to determine the cohomology of moduli spaces of Higgs bundles on any given curve (in the coprime situation). As an application of the method we compute the y-genus of the space of PGL_n-Higgs bundles for any rank n, confirming a conjecture of T. Hausel.
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is proven that the moduli space of stable rank-2 vector bundles with Chern classes c_1=1,c_2=5 on a generic X is isomorphic to the curve of genus 7 obtained by taking an orthogonal linear section of the spinor tenfold. This is an inverse of Mukai's result on the isomorphism of a non-abelian Brill--Noether locus on a curve of genus 7 to a Fano threefold of genus 7. An explicit geometric construction of both isomorphisms and a similar result f…
Let C be a curve of genus g, and let SU(r) be the moduli space of vector bundles of rank r on C, with trivial determinant. A general E in SU(r) defines a theta divisor in the linear system |r Theta|, where Theta is the canonical theta divisor in Pic^{g-1}(C). This defines a rational map SU(r) - - > |r Theta|, which is the map associated to the determinant bundle on SU(r) (the positive generator of Pic(SU(r)). In this paper we prove that in genus 2 this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta…
We give an explicit expression of the Hitchin Hamiltonian system for rank two vector bundles with trivial determinant bundle over a curve of genus two.
Let SU_X(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P^8 and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P^8 there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that these two hypersurfaces are dual. As an immediate corollary, we derive a Torelli-type result.
Let X be a general complex Fano threefold of genus 8. We prove that the moduli space of rank two semistable sheaves on X with Chern numbers c_1=1, c_2=6 and c_3=0 is isomorphic to the Fano surface F(X) of conics on X. Inside F(X), the non-locally free sheaves are parameterized by a smooth curve of genus 26 isomorphic to the base of the family of lines on X.
Let $X$ be a smooth projective curve with genus $g\geq3$. Let $\mathcal{N}$ be the moduli space of stable rank two vector bundles on $X$ with a fixed determinant $\mathcal{O}_X(-x)$ for $x\in X$. In this paper, as a generalization of Kiem and Castravet's works, we study the stable maps in $\mathcal{N}$ with genus $0$ and degree $3$. Let $P$ be a natural closed subvariety of $\mathcal{N}$ which parametrizes stable vector bundles with a fixed subbundle $L^{-1}(-x)$ for a line bundle $L$ on $X$. We describe the stable map space $\mathbf{M}_0(P,3)$. It turns out that the space $\mathbf{M}_0(P,3)$ …
Given a complex curve C of genus 2, there is a well-known relationship between the moduli space of rank 3 semistable bundles on C and a cubic hypersurface known as the Coble cubic. Some of the aspects of this is known to be related to the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space. We extend this relationship to arbitrary fields and study some of the connections to invariant theory, which will be studied more in-depth in a followup paper.
The moduli space M_2 of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 = 1-g over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Ray…
Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\leq p \leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double étale …
Let $X$ be a smooth proper genus 2 curve over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the the moduli space $M\_X$ of semi-stable rank 2 vector bundles over $X$, which is isomorphic to a 3-dimensional projective space. Y. Laszlo and C. Pauly recently gave the equations of $F$ for an ordinary $X$. Using deformation, we give these equations for a supersingular $X$ and draw some consequences such as the base locus of $F$ (one point), or the stability of the complementary Zariski open set.
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve G, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of G. It is proven that, for a generic X, the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c_1=-K_X and deg c_2=6 and that the latter is birational to the singular locus of the theta divisor of J(X).
In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.
Let $C\subset \mathbb {P}^n$ be a smooth curve and $N_C$ its normal bundle. $N_C$ satisfies strong interpolation if for all integers $s>0$ and $λ_i\in \{0,1,\dots ,n-1\}$, $1\le i \le s$, there are distinct points $P_1,\dots ,P_s\in C$ and linear subspaces $U_i\subseteq E|P_i$ such that $\dim (U_i)= λ_i$ for all $i$ and the evaluation map $H^0(E)\to \oplus _{i=1}^{s} U_i$ has maximal rank (A. Atanasios). We prove that $C$ satisfies strong interpolation if either $C$ is a linearly normal elliptic curve or $C$ is a general embedding of degree $d\ge (5n-8)g+2n^2-5n+4$ of a smooth curve $X$ of gen…
A generic strictly semistable bundle of degree zero over a curve X has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For X of genus at least 5, we construct stable vector bundles over X of rank $r$ for all $r \geq 5$, with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the appendix, Raynaud's original example of a stable rank 2 vector bundle with reducibl…
Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a canonical theta divisor $Θ$. The space $SU_C(3)$ is a double cover of $P^8=|3Θ|$ branched along a sextic hypersurface, the Coble sextic. In the dual $\check{P}^8=|3Θ|^*$, where $J^1$ is embedded, there is a unique cubic hypersurface singular along $J^1$, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. More…
We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy around the singular fibers is unipotent. This relative canonical bundle is the (canonical extension of) the Hodge bundle and the inequality is generalized to the degrees of the Hodge bundles of a complex variation of Hodge structures.
We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of ${\mathbb C}^*$-actions on semiprojective varieties, ${\mathbb C}^*$ characters of indices of ${\mathbb C}^*$-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier-Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscul…
Given a smooth prime Fano threefold $X$ of genus 7 we consider its homologically projectively dual curve $Γ$ and the natural integral functor $Φ^{!}:D^b(X) \to D^b(Γ)$. We prove that, for $d\geq 6$, $Φ^{!}$ gives a birational map from a component of the moduli scheme $M_X(2,1,d)$ of rank 2 stable sheaves on $X$ with $c_1=1$, $c_2=d$ to a generically smooth $2d-9$-dimensional component of the Brill-Noether variety $W^{2d-11}_{d-5,5d-24}$ of stable vector bundles on $Γ$ of rank $d-5$ and degree $5d-24$ with at least $2d-10$ sections. This map turns out to be an isomorphism for $d=6$, and the mod…
Let $X$ be genus 2 curve defined over an algebraically closed field of characteristic $p$ and let $X\_1$ be its $p$-twist. Let $M\_X$ (resp. $M\_{X\_1}$) be the (coarse) moduli space of semi-stable rank 2 vector bundles with trivial determinant over $X$ (resp. $X\_1$). The moduli space $M\_X$ is isomorphic to the 3-dimensional projective space and is endowed with an action of the group $J[2]$ of order 2 line bundles over $X$. When $3\leq p \leq 7$, we show that the Verschiebung (i.e., the separable part of the action of Frobenius by pull-back) $V : M\_{X\_1} \dashrightarrow M\_X$ is completely…
We study stable rank 2 vector bundles with trivial determinant whose Frobenius pull back is non stable over a general curve of genus g>1. In genus 2, we apply recent results about the theta divisor associated to the bundle B of locally exact differential forms and we derive a scheme-theoretic description of the Frobenius inverse image of the semi-stable boundary of the moduli space.
We compute the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least three over the complex numbers. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles, when the genus is at least four.
In this paper we study the formal classication of two-dimensional neighborhoods of genus g $\ge$ 2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal/analytic classication of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.
A new genus $g=g(X,\ce)$ is defined for the pairs $(X,\ce)$ that consist of $n$-dimensional compact complex manifolds $X$ and ample vector bundles $\ce$ of rank $r$ less than $n$ on $X$. In case $r=n-1$, $g$ is equal to curve genus. Above pairs $(X,\ce)$ with $g$ less than two are classified. For spanned $\ce$ it is shown that $g$ is greater than or equal to the irregularity of $X$, and its equality condition is given.
We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4-manifold of non-zero signature which is an oriented fiber bundle over a Riemann surface. A refined version of the problem asks for the minimal base genus for fixed signature and fiber genus. Our constructions also provide new (asymptotic) upper bounds for these numbers.
This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or mo…
Raynaud has shown that over a general curve of genus $g \ge 2$, every semistable bundle of rank three and integral slope admits a theta divisor. We show that this can fail for special curves: Over any bielliptic curve of genus $g \ge 5$, we construct a stable rank three bundle of trivial determinant with no theta divisor. This gives a partial answer to a question of Beauville.
In this paper we investigate generalized theta divisors $Θ_r$ in the moduli spaces $\mathcal{U}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $Φ$ of $Θ_r$ in terms of a projective bundle $π:\mathbb{P}(\mathcal{V})\to\mathcal{U}_C(r-1,r)$ which parametrizes extensions of stable vector bundles on the base by $\mathcal{O}_C$. Then, we study the composition of $Φ$ with the well known theta map $θ$. We prove that, when it is restricted to the general fiber of $π$, we obtain a linear embedding.
Any line bundle $\cl $ on a smooth curve $C$ of genus $g$ with $°\cl \ge 2g+1$ is normally generated, i.e., $φ_\cl (C)\subseteq \mathbb P H^0 (C,\cl)$ is projectively normal. However, it has known that more various line bundles of degree $d$ failing to be normally generated appear on multiple coverings of genus $g$ as $d$ becomes smaller than $2g+1$. Thus, investigating the normal generation of line bundles on multiple coverings can be an effective approach to the normal generation. In this paper, we obtain conditions for line bundles on multiple coverings being normally generated or not, resp…
The purpose of this note is to extend Beilinson and Drinfeld's "very good" property to moduli stacks of parabolic vector bundles on curves of genuses $g = 0$ and $g = 1$. Beilinson and Drinfeld show that for $g > 1$ a trivial parabolic structure is sufficient for the moduli stacks to be "very good". We give a necessary and sufficient condition on the parabolic structure for this property to hold in the case of lower genus.
We study a generalization of Lian-Liu-Yau's notion of Euler data in genus zero and show that certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli with markings induce data satisfying the generalization. In the case of one or two markings, this data is explicitly identified in terms of hypergeometric type classes, constituting a complete extension of Lian-Liu-Yau's mirror principle in genus zero to the case of two marked points and establishing a program for the general case. We give several applications involving the Euler class of obstructio…
This note is an attempt to generalize Bolibruch's theorem from the projective line to curves of higher genus. We show that an irreducible representation of the fundamental group of an open in a curve of higher genus has always a representation as a regular system of differential equations on a semistable bundle of degree 0. Vice-versa, we show that given such a bundle and 3 points on the curve, one can construct an irreducible representation of the curve minus the 3 points such that an associated regular system of differential equations lives on this bundle.
Let $X$ be a compact complex manifold of dimension $n\ge 2$ and $\ce$ an ample vector bundle of rank $r<n$ on $X$. As the continuation of Part I, we further study the properties of $g(X,\ce)$ that is an invariant for pairs $(X,\ce)$ and is equal to curve genus when $r=n-1$. Main results are the classifications of $(X,\ce)$ with $g(X,\ce)=2$ (resp. 3) when $\ce$ has a regular section (resp. $\ce$ is ample and spanned) and $1<r<n-1$.
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
We study the moduli space of trace-free irreducible rank 2 holomorphic connections over a complex projective curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus $2$ case, connections as above are invariant under the hyperelliptic involution : they descend as rank $2$ logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical a…
We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the …
Two definitions of the Clifford index for vector bundles on a smooth projective curve $C$ of genus $g\ge4$ were introduced in a previous paper by the authors. In another paper the authors obtained results on one of these indices for bundles of rank 3. In this paper we extend these results in the case where $C$ has classical Clifford index 3. In particular we prove Mercat's conjecture for bundles of rank 3 for $g \leq 8$ and $g \geq 13$ when $C$ has classical Clifford index 3. We obtain complete results in the case of genus 7.
The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $Γ$, the associated canonical linear system $\vert K_Γ\vert$ has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.
