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Given any compact Riemann surface C, there is a canonical meromorphic 2–form η ˆ on C × C , with pole of order two on the diagonal Δ ⊂ C × C , constructed in [4] . This meromorphic 2–form η ˆ produces a canonical projective structure on C. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface C. We prove that these two projective structures differ in general. This is done by comparing the ( 0 , 1 ) –component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The ( 0 , 1 ) –component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in [16] as the Weil–Petersson Kahler form ω w p on the moduli space of curves. We prove that the ( 0 , 1 ) –component of the differential of the section of the moduli space of projective structures corresponding to η ˆ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.

In this paper we study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure (IVHS) is of rank 1. We show that if the genus g is greater or equal to 8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete a result of Naranjo and Pirola. We show in fact that if the genus g is greater or equal to 6, the hyperelliptic locus is the only 2g-1-dimensional sub-locus Y of the moduli space of curves of genus g, such that for the general element [C] in Y, its Jacobian J(C) is dominated by a hyperelliptic Jacobian of genus g' greater or equal to g., Comment: 10 pages

Let $$f:S\rightarrow B$$ be a fibration of curves and let $$f_*\omega _{S/B}={{\mathcal {U}}}\oplus {{\mathcal {A}}}$$ be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $${{\mathcal {U}}}$$. The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole $${{\mathcal {U}}}$$ of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).

This paper contains two results on Hodge loci in the moduli space of curves. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fibers is contained in a proper Hodge locus. The second result deals with divisors in the moduli space of curves. It is proved that the image of a divisor in the moduli of principally polarized abelian varieties is not contained in a proper totally geodesic subvariety. It follows that a Hodge locus in the moduli space of curves has codimension at least 2., To appear on Journal de l'Institut de Math\'ematiques de Jussieu

Motivated by results of Mestre and Voisin, in this note we mainly consider abelian varieties isogenous to hyperelliptic Jacobians In the first part we prove that a very general hyperelliptic Jacobian of genus $g\ge 4$ is not isogenous to a non-hyperelliptic Jacobian. As a consequence we obtain that the Intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary tells that the Jacobian of a very general $d$-gonal curve of genus $g \ge 4$ is not isogenous to a different Jacobian. In the second part we consider a closed subvariety $\mathcal Y \subset \mathcal A_g$ of the moduli space of principally polarized varieties of dimension $g\ge 3$. We show that if a very general element of $\mathcal Y$ is dominated by a hyperelliptic Jacobian, then $\dim \mathcal Y\ge 2g$. In particular, if the general element in $\mathcal Y$ is simple, its Kummer variety does not contain rational curves. Finally we show that a closed subvariety $\mathcal Y\subset \mathcal M_g$ of dimension $2g-1$ such that the Jacobian of a very general element of $\mathcal Y$ is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus., Comment: New version. Accepted in Adavances in Mathematics

Let $X$ be an irreducible projective variety and $f$ a morphism $X \rightarrow \mathbb{P}^n$. We give a new proof of the fact that the preimage of any linear variety of dimension $k\ge n+1-\dim f(X)$ is connected. We prove that the statement is a consequence of the Generalized Hodge Index Theorem using easy numerical arguments that hold in any characteristic. We also prove the connectedness Theorem of Fulton and Hansen as application of our main theorem., Comment: 11 pages, New version, incorporating the referee's suggestions

This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $g\geq 7$ and $g'\geq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve., 13 pages; exposition improved; Ann. Sc. Norm. Super. Pisa Cl. Sci. (to appear)

In this paper, we study totally geodesic subvarieties [Formula: see text] of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for [Formula: see text]. We prove that if [Formula: see text] is generically contained in the Torelli locus, then [Formula: see text].

In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory., 11 pages, final version to appear in BLMS

We construct three new families of fibrations $\pi : S \to B$ where $S$ is an algebraic complex surface and $B$ a curve that violate Xiao's conjecture relating the relative irregularity and the genus of the general fiber. The fibers of $\pi$ are certain \'etale cyclic covers of hyperelliptic curves that give coverings of $P^1$ with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill-Noether range., Comment: 12 pages. Exposition improved. The last section has been expanded with more details of the proof. Accepted for publication in Ann. Mat. Pura Appl

In this note we prove a sharp bound for the number of 2-torsion points on a theta divisor and show that this is achieved only in the case of products of elliptic curves. This settles in the affirmative a conjecture of Marcucci and Pirola.

We prove a new inequality for the Hodge number $$h^{1,1}$$ of irregular complex smooth projective surfaces of general type without irrational pencils of genus $$\ge $$ 2. More specifically we show that if the irregularity $$q$$ satisfies $$q=2^k+1$$ then $$h^{1,1}\ge 4q-3$$ . This generalizes results previously known for $$q=3$$ and $$q=5$$ .

We study the components of the Hurwitz scheme of ramified coverings of $\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the $3$-cycles, we use as invariant the lifting to the Valentiner group, triple covering of $A_6$. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution found by Bogomolov and Kulikov., Comment: 18 pages

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to the two-dimensional case and we construct a new family of surfaces which are Lagrangian in their Albanese variety. Moreover, we analyze these surfaces computing their Chern invariants, and proving that they are not fibred over curves of genus greater than one., Comment: 36 pages, 3 figures

We study the Saito-Ikeda infinitesimal invariant of the ccle defined by curves in their jacobians using rank (k+1) vector bundles and we give a criterion for which the higher cycle class map is not trivial. When k=2, this turns out to be strictly linked to the Petri map for vector bundles: an explicit construction on a curve of genus g>9 shows the existence of a non trivial element in the higher Griffiths group., 17 pages; accepted for pubblication in Nagoya Math. Journal

We prove by means of the study of the infinitesimal variation of Hodge structure and a generalization of the classical Babbage-Enriques-Petri theorem that the Jacobian variety of a generic element of a $k$ codimensional subvariety of $\mathcal M_g$ is not isogenous to a distinct Jacobian if $g>3k+4$. We extend this result to $k=1, g\ge 5$ by using degeneration methods.

The uniform position principle states that, given an irreducible nondegenerate curve C in the projective r-space $P^r$, a general (r-2)-plane L is uniform, that is, projection from L induces a rational map from C to $P^1$ whose monodromy group is the full symmetric group. In this paper we show the locus of non-uniform (r-2)-planes has codimension at least two in the Grassmannian for a curve C with arbitrary singularities. This result is optimal in $P^2$. For a smooth curve C in $P^3$ that is not a rational curve of degree three, four or six, we show any irreducible surface of non-uniform lines is a Schubert cycle of lines through a point $x$, such that projection from $x$ is not a birational map of $C$ onto its image., corrected typo in first paragraph of introduction, 23 pages, AMSLaTeX

Let X X be a compact Riemann surface of genus g g and d ≥ 12 g + 4 d\geq 12g+4 be an integer. We show that X X admits meromorphic functions with monodromy group equal to the alternating group A d . A_d.

We study the complete minimal surfaces of bounded total (Gaussian) curvature in the Euclidean space. By the fundamental result of Osserman (see [12], [13]) if F : Y → R is such a minimal immersion then Y is conformally equivalent to a compact Riemann surface X minus a finite number of points and the Weierstrass data (see 0.19 below) extends meromorphically to X. A natural question ([13], page 151) is to give more examples of such surfaces. In [6] it is proved that for any connected compact Riemann surface X there is a finite set of points Z = {p1, . . . , pn} and a minimal immersion F : X − Z → R such that F (X − Z) is complete, non rigid and of finite total curvature. Many more examples were found by Yang (cf. [15]) where a bound on the degree of Z is given. The non rigid minimal surfaces of finite curvature are algebraic geometric objects. In fact a theorem of Calabi implies (cf. [4] and see 0.16, 0.27) that F is the real part of an algebraic morphism Φ : X − Z → C. To show our theorem we adopt the spinor viewpoint of [10]. In our opinion it has several advantages. Firstly (see 1.8) the components which correspond to different spin bundles are separated. Then the existence result follows by a Chern class computation. We notice the analogy with the Brill-Noether theory (cf. [1]). This is the easy part of the job. The duty is to prove that some of those surfaces are indeed immersed. The spinor representation allows to compute the holomorphic tangent space of the moduli variety, say F , representing them. In fact, as is neatly explained in [10], some quadratic equations become linear. We prove, under suitable hypothesis on the polar divisor, that a component of F , say G, has the

In this paper we investigate the existence of generically finite dominant rational maps from products of curves to surfaces of general type. We prove that the product CxD of two distinct very general curves of genus g>6 and g'>1 does not admit dominant rational maps on other surfaces of general type., 13 pages, minor changes, to appear on Bull. Lond. Math. Soc

We give a bound on the number of points of order two on the theta divisor of a principally polarized abelian variety A. When A is the Jacobian of a curve C the result can be applied in estimating the number of effective square roots of a fixed line bundle on C., Comment: 4 pages

In this paper a new intrinsic geometric characterization of the symmetric square of a curve and of the ordinary product of two curves is given. More precisely it is shown that the existence on a surface of general type S of irregularity q of an effective divisor D having self-intersection D^2>0 and arithmetic genus q implies that S is either birational to a product of curves or to the second symmetric product of a curve., International Mathematics Research Notices 2011

The general topological framework for testing the possible occurrence of anomalies in gauge theories can be constructed in terms of the theory of group actions on line bundles through the introduction of a suitable group cohomology. In this Letter, we generalize this construction in such a way that it can be applied to a larger class of theories, allowing for a noncontractible configuration space and a nonconnected ‘gauge’ group. This construction find applications to the problem of the lifts of principal group actions. As a physical application, we compare the mechanisms of the anomalies cancelation in gauge and string theories, through a geometrical splitting of local and global anomalies.

We give a bound for the number of rational maps between algebraic varieties of general type under mild hypothesis on the canonical map. We use an idea inspired by Tanabe's work. Instead of attaching a morphism of Hodge structures to a rational map we simply associate to it a piece of the integral Hodge lattice. This procedure does not give an injective map, but by means of a geometric argument, we can estimate the number of maps with the same image.

This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call \textit{locally toric} a projective curve that in a neighbourhood of every point has a local analytical parametrization of type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in $\bP^3$ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \cite{kaji} by H.Kaji where the same result is proven for any curve in $\bP^3$ such that every branch is smooth. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations $(t^{a_1},...,t^{a_n})$, the general osculating 2-space to a locally toric curve of genus $g, 23 pages

We calculate the maximal dimension of linear spaces of symmetric and hermitian matrices with given high rank, generalizing a well-known result of Adams {\em et al.}, Comment: 7 pages

Let X be a smooth complex projective surface. We prove that for any sufficiently big m there exists a rational dominant map f from X into a complex rational ruled surface Y, such that f is generically finite of degree m and has monodromy the alternating group Am., Comment: 17 pages

In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorphic manner., Comment: 10 pages

We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k, such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kaehler-Einstein metric too., 30 pages, Latex

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto φ(C) ⊂ Pr. We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H0(C, L2) can be written as α2 + β2 + γ2, with α, β, γ ∈ H0(C, L). If there are no quadrics of rank 3 containing φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem. Copyright © 2004 by Marcel Dekker, Inc.

The aim of this paper is to construct non-trivial cycles in the first higher Chow group of the Jacobian of a curve having special torsion points. The basic tool is to compute the analogue of the Griffiths' infinitesimal invariant of the natural normal function defined by the cycle as the curve moves in the corresponding moduli space. We prove also a Torelli-like theorem. The case of genus 2 is considered in the last section.

Zariski proved the general complex projective curve of genus g>6 is not rationally uniformized by radicals, that is, admits no map to the projective line whose Galois group is solvable. We give an example of a genus 7 complex projective curve Z that is not rationally uniformized by radicals, but such that there is a finite covering Z' -> Z with Z' rationally uniformized by radicals. The curve providing the example appears in a paper by Debarre and Fahlaoui where a construction is given to show the Brill Noether loci W_d(C) in the Jacobian of a curve C may contain translates of abelian subvarieties not arising from maps from C to other curves., 8 pages, AMSlatex

Our aim is to compute the infinitesimal invariant of a normal function associated to a very natural cycle: the curve in its Jacobian. We have to solve an exercise along some lines explained on these Cime lectures. This “exercise” is a joint work with Alberto Collino [CP] with a big hint given by F. Bardelli and S. Muller Stach that we will explain below.

Si studiano le classi di Chern delle varieta determinantali in ipotesi di buone codimensioni. Si esplicitano in particolare i primi termini del carattere di Chern di tali varieta. Applicando tali risultati alle varieta W d r dei divisori speciali su una curva algebrica di tipo generale quando dim (W d r )=1,se ne calcula il genere.