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It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g., 18 pages. Minor improvements, also referencing the related work of Lian. Final version to appear in Annali Scuola Normale di Pisa

Let $$F:{\mathcal {V}}\rightarrow B$$ F : V → B be a smooth non-isotrivial $$1-$$ 1 - dimensional family of complex polarized abelian varieties and $$V_b=F^{-1}(b)$$ V b = F - 1 ( b ) be the general fiber. Let $${\mathcal {F}}^1\subset R^1F_*{\mathbb {C}}$$ F 1 ⊂ R 1 F ∗ C be the associated Hodge bundle filtration, $${\mathcal {F}}^1_b=H^{1.0}(V_b).$$ F b 1 = H 1.0 ( V b ) . Under the assumption that the Fujita decomposition for $${\mathcal {F}}^1$$ F 1 is non trivial, that is there is a non trivial flat sub-bundle $$0\ne {\mathbb {U}}\subset {\mathcal {F}}^1,$$ 0 ≠ U ⊂ F 1 , we show that $$V_b$$ V b has non-trivial endomorphism: $$End(V_b)\ne {\mathbb {Z}}.$$ E n d ( V b ) ≠ Z .

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.

We study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { x } - { 0 A } is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0, Comment: 9 pages

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).

Let S be a smooth algebraic surface in $${\mathbb {P}}^3({\mathbb {C}})$$ P 3 ( C ) . Movasati and Sertöz (Rend. Circ. Mat. Palermo 2:1–17, 2020) associate an ideal $$I_{\alpha (C)}$$ I α ( C ) to the primitive cohomology class $$\alpha (C)$$ α ( C ) of C in S. We show that the equations of C can be determined by $$I_{\alpha (C)}$$ I α ( C ) under numerical conditions. We apply this result to reconstruct rational curves and arithmetically Cohen-Macaulay curves from their cohomology classes. On the other hand, we show that the class $$\alpha (C)$$ α ( C ) of a rational quartic curve C on a smooth quartic surface S is not even perfect, that is, that $$I_{\alpha (C)}$$ I α ( C ) is bigger than the sum of the Jacobian ideal of S and of the homogeneous ideals of curves D in S for which $$I_{\alpha (D)}=I_{\alpha (C)}$$ I α ( D ) = I α ( C ) .

In this paper we introduce some {\it variation functions} associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if $X$ is a smooth plane curve $X$, then there exists a first order deformation $\xi\in H^1(T_X)$ which deforms $X$ as plane curve, such that $\xi\cdot:H^0(\omega_X)\to H^1(\mathcal{O}_{X})$ is an isomorphism. We also generalize the notions of variation functions to the higher dimensional case and we analyze the link between IVHS and the Weak and Strong Lefschetz properties of the Jacobian ring of a smooth hypersurface., Comment: 19 pages. Comments are welcome

The Quillen connection on ${\mathcal L} \rightarrow {\mathcal M}_g$, where ${\mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${\mathcal M}_g$, $g \geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${\mathcal M}_g$. The bundle of holomorphic connections on ${\mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${\mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^\infty$ section of the first bundle given by the Quillen connection on ${\mathcal L}$ to the $C^\infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.

We investigate a class of odd (ramification) coverings $C \to \mathbb{P}^1$ where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of $\mathbb{P}^1$ and the covering map makes the hyperelliptic involution of $C$ commute with an involution of $\mathbb{P}^1$. We show that the total number of hyperelliptic odd coverings of minimal degree $4g$ is ${3g \choose g-1} 2^{2g}$ when $C$ is general. Our study is approached from three main perspectives: if a fixed effective theta characteristic is fixed they are described as a solution of a certain class of differential equations; then they are studied from the monodromy viewpoint and a deformation argument that leads to the final computation., Comment: 15 pages, comments very welcome

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

In this paper we give a geometric interpretation of the second fundamental form of the period map of curves and we use it to improve the upper bounds on the dimension of a totally geodesic subvariety Y of A_g generically contained in the Torelli locus obtained in [3], [7]. We get dim Y < 2g if g is even, dim Y < 2g+1 if g is odd. We also study totally geodesic subvarieties Z of A_g generically contained in the hyperelliptic Torelli locus and we show that dim Z < g+2., Final version. To appear in Proceedings of the AMS

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].

Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that the relative irregularity of $f$, $q_f=h^{1,0}(S)-b$ is less than or equal to $g(F)-c_f$. In particular this proves the (modified) Xiao's conjecture: $q_f\le 1+g(F)/2$ for fibrations of general Clifford index. In this short note we assume that the generic fiber of $f$ is a plane curve of degree $d\ge 5$ and we prove that $q_f\le g(F)-c_f-1$. In particular we obtain the conjecture for families of quintic plane curves. This theorem is implied for the following result on infinitesimal deformations: let $F$ a smooth plane curve of degree $d\ge 5$ and let $\xi$ be an infinitesimal deformation of $F$ preserving the planarity of the curve. Then the rank of the cup-product map $\cdot \xi: H^0(F,\omega_F) \rightarrow H^1(F,O_F)$ is at least $d-3$. We also show that this bound is sharp., Comment: 8 pages. Some typos have been corrected. To appear in "Geometriae Dedicata"

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

Let $C$ be a smooth curve which is complete intersection of a quadric and a degree $k>2$ surface in $\mathbb{P}^3$ and let $C^{(2)}$ be its second symmetric power. In this paper we study the finite generation of the extended canonical ring $R(\Delta,K) := \bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},a\Delta+bK)$, where $\Delta$ is the image of the diagonal and $K$ is the canonical divisor. We first show that $R(\Delta,K)$ is finitely generated if and only if the difference of the two $g_k^1$ on $C$ is torsion non-trivial and then show that this holds on an analytically dense locus of the moduli space of such curves., Comment: 16 pages

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 $$q=5$$ .

For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number \rho(C, r) and we prove, under some mild additional assumptions, that if \rho(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to \rho(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly on a smooth projective variety X of maximal Albanese dimension and inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension.

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 dominant rational maps from a general surface in to surfaces of general type. We prove restrictions on the target surfaces, and special properties of these rational maps. We show that for small degree the general surface has no such map. Moreover a slight improvement of a result of Catanese, on the number of moduli of a surface of general type, is also obtained.

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

Let $E$ be a vector bundle on a smooth complex projective variety $X$. We study the family of sections $s_t\in H^0(E\otimes L_t)$ where $L_t\in Pic^0(X)$ is a family of topologically trivial line bundle and $L_0=\mathcal O_X,$ that is, we study deformations of $s=s_0$. By applying the approximation theorem of Artin [2] we give a transversality condition that generalizes the semi-regularity of an effective Cartier divisor. Moreover, we obtain another proof of the Severi-Kodaira-Spencer theorem [4]. We apply our results to give a lower bound to the continuous rank of a vector bundle as defined by Miguel Barja [3] and a proof of a piece of the generic vanishing theorems [6] and [7] for the canonical bundle. We extend also to higher dimension a result given in [8] on the base locus of the paracanonical base locus for surfaces., 12 pages. An extra hypothesis is added to the results in last section. Final version will appear in Collectanea Mathematica

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.

We give some upper bounds on the dimension of the kernel of the cup product map \(H^{1}(X,\mathbb{C}) \otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})\), where X is a compact Kahler variety without Albanese fibrations.

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.

In this paper it is proved that a complete algebraic surface of general type with p g =q=3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p g =q=3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface.

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

Given a smooth complex projective variety [math] , a line bundle [math] of [math] and [math] , we say that [math] is [math] –transversal to [math] if the complex [math] is exact. We prove that if [math] is [math] –transversal to [math] and [math] satisfies [math] , then the first order deformation [math] of the pair [math] in the direction [math] extends to an analytic deformation. ¶ We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld and Popa in higher dimension. In particular, we prove the inequality [math] for a variety [math] of maximal Albanese dimension without irregular fibrations of Albanese general type.

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

In a previous paper, we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing the natural parametrization of maps by means of the space of linear projections in a suitable projective space, and this leads to some new insight in the geometry of the finiteness theorem., Comment: 9 pages, to appear in Proc. Amer. 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

We carry out an analysis of the canonical system of a minimal complex surface of general type with irregularity q>0. Using this analysis we are able to sharpen in the case q>0 the well known Castelnuovo inequality K^2>=3p_g+q-7. Then we turn to the study of surfaces with p_g=2q-3 and no fibration onto a curve of genus >1. We prove that for q>=6 the canonical map is birational. Combining this result with the analysis of the canonical system, we also prove the inequality: K^2>=7\chi+2. This improves an earlier result of the first and second author [M.Mendes Lopes and R.Pardini, On surfaces with p_g=2q-3, Adv. in Geom. 10 (3) (2010), 549-555]., Comment: Dedicated to Fabrizio Catanese on the occasion of his 60th birthday. To appear in the special issue of Science of China Ser.A: Mathematics dedicated to him. V2:some typos have been corrected

Let A A be an n n -dimensional Abelian variety, n ≥ 2 n \geq 2 ; let CH 0 ( A ) {\text {CH}_0}(A) be the group of zero-cycles of A A , modulo rational equivalence; by regarding an effective, degree k k , zero-cycle, as a point on S k ( A ) {S^k}(A) (the k k -symmetric product of A A ), and by considering the associated rational equivalence class, we get a map γ : S k ( A ) → CH 0 ( A ) \gamma :{S^k}(A) \to {\text {CH}_0}(A) , whose fibres are called γ \gamma -orbits. For any n ≥ 2 n \geq 2 , in this paper we determine the maximal dimension of the γ \gamma -orbits when k = 2 k = 2 or 3 3 (it is, respectively, 1 1 and 2 2 ), and the maximal dimension of families of γ \gamma -orbits; moreover, for generic A A , we get some refinements and in particular we show that if dim ⁡ ( A ) ≥ 4 \dim (A) \geq 4 , S 3 ( A ) {S^3}(A) does not contain any γ \gamma -orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: A t = E t × B {A_t} = {E_t} \times B ( E t {E_t} is an elliptic curve with varying moduli) and we have constructed suitable projections between S k ( A t ) {S^k}({A_t}) and S k ( B ) {S^k}(B) which preserve the dimensions of the families of γ \gamma -orbits; then we have done induction on n n . For n = 2 n = 2 the proof is based upon the papers of Mumford and Roitman on this topic.

In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and g>5. We also show that a very generic Prym variety of dimension at least 4 is not isogenous to a Jacobian., 24 pages, 2 figures. One reference added. To appear in Compositio Mathematica