We prove that if $X$ and $S$ are smooth varieties and $f\colon X\to S$ is an elliptic fibration with singular fibers curves of types I$_N$ with $N\geq 1$, II, III and IV, then the relative Jacobian $\hat{f}\colon \bar{M}_{X/S}\to S$ of $f$, defined as the relative moduli space of semistable pure dimension one sheaves of rank 1 and degree 0 on the fibers of $f$, is an elliptic fibration such that all its fibers are irreducible. This extends known results when fibers are integral or of type I$_2$.
In this paper we give the Weierstrass equations and the generators of Mordell-Weil groups for Jacobian fibrations on the singular K3 surface of discriminant 3.
In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of P^2 ramified along six lines in general position.
In this paper we give the Weierstrass equations for Jacobian fibrations on the K3 surface that is the minimal resolution of the double covering of projective plane ramified along generic six lines.
For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification of the intermediate Jacobian fibration giving a ten-dimensional compact hyper-Kähler manifold, which we then show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Markushevich.
The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.
For any projective curve $X$ let $\bar{M}^d(X)$ be the Simpson moduli space of pure dimension one rank 1 degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. When $X$ is a reduced curve the connected component of $\bar{M}^d(X)$ that contains semistable line bundles can be considered as the compactified Jacobian of $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian for the following projective curves: tree-like curves and all reduced and reducible curves that can appear as Kodaira singular fibers of an elliptic fibration,…
We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
Let Y->P^n be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that X is a Beauville-Mukai integrable system if n=3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville-Mukai system if n=3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.
(1,d)-polarized abelian surfaces in P^(d-1) with two plane cubic curve fibrations lie in two elliptic P^2-scrolls. The union of these scrolls form a reducible Calabi-Yau 3-fold. In this paper we show that this occurs when d<10 and analyse the family of such surfaces and 3-folds in detail when d=6. In particular, the reducible Calabi-Yau 3-folds deform in that case to irreducible ones with non-normal singularities.
Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20.
L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow \mathbb{P}^1$ of genus 2 curves with supersingular jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.
Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. Here we further identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.
We describe all the elliptic models with section on the Shioda supersingular K3 surface X of Artin invariant 1 over an algebraically closed field of characteristic 3. In particular, we compute elliptic parameters and Weierstrass equations for the fifty two different fibrations, and analyze the reducible fibers and Mordell-Weil lattices.
We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptiona…
A Kodaira fibration is a non-isotrivial fibration $f\colon S\rightarrow B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space $\mathcal{M}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of $\mathcal{M}_3$ parametrizing curves whose…
We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on $R_{5,5}$. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in de…
In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of its Jacobian.
We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of the Jacobian of a curve associated to the fibration. We remark that this is related to Recillas' trigonal construction. Finally we discuss the two-torsion in the Brauer group of a general K3 surface with a polarisation of degree two.
We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the family of those curves D in the surface which are flat over the base of the elliptic fibration. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of sheaves whose restric…
For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2λ_1+2λ_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution $\tilde M$, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of $\tilde M$ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin arXiv:161…
We investigate the Shioda-Inose structure of the Jacobian of a smooth complex genus-2 curve C arising from its degree-2 elliptic subcovers and determine the Mordell-Weil groups and lattices in the case of a semistable fibration having exactly four singular fibers.
Consider a smooth projective surface $S$. Consider a fibration $S\to C$ where $C$ is a quasi-projective curve such the fibers are smooth projective curves. The aim of this text is to show that the kernels of the push-forward homomorphism $\{j_{t*}\}_{t\in C}$ from the Jacobian $J(C_t)$ to $A_0(S)$ forms a family in the sense that it is a countable union of translates of an abelian scheme over $C$ sitting inside the Jacobian scheme $\mathscr{J}\to C$, such that the fiber of this countable union at $t$ is the kernel of $j_{t*}$.
We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi-Yau threefolds we show that the Fourier-Mukai transform induces an embedding ion of the relative Jacobian of spectral line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves to a generic torus fibration over the moduli space of curves of given arithmetic genus on the Calabi-Yau manifold.
We describe explicit formulas relevant to the F-theory/heterotic string duality that reconstruct from a specific Jacobian elliptic fibration on the Shioda-Inose surface covering a generic Kummer surface the corresponding genus-two curve using the level-two Satake coordinate functions. We derive explicitly the rational map on the moduli space of genus-two curves realizing the algebraic correspondence between a sextic curve and its Satake sextic. We will prove that it is not the original sextic defining the genus-two curve, but its corresponding Satake sextic which is manifest in the F-theory mo…
Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau $d$-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polyto…
This thesis, done under the supervision of Filippo Viviani, is devoted to the study of the spectral correspondence for $G$-Higgs pairs, in the case of $G=SL(r,\mathbb{C})$, $PGL(r, \mathbb{C})$, $Sp(2r,\mathbb{C})$, $GSp(2r,\mathbb{C})$, $PSp(2r,\mathbb{C})$, over any fiber. In the first chapter we introduce the compactified Jacobian stack parametrizing torsion-free rank-1 sheaves over a (possibly reducible, non reduced) projective curve and we describe the spectral correspondence for Higgs pairs in terms of the compactified Jacobian. In the second chapter, we study the notion of direct and in…
We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group of order 11, $11\rtimes 5$, $L_2(11)$ and the Mathieu groups $M_{11}$, $M_{22}$. We also show that a surface $X$ admitting an automorphism $g$ of order 11 admits a $g$-invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces.
