Hyperelliptic Jacobians and isogenies

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 Mathematic

Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane

It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length.Comment: v2: Minor changes after journal review. This text uses the theory developed in arXiv:1304.1788 for the specific example of a homogeneous ellipsoid rolling on the plan

Walking down the borderline: hybridity and the modulation of the self in three canonical chicano novels

This paper presents an analysis of three Chicano novels considered canonical, in which the concept of “hybridity” is discussed in relation with the individual: Rudolfo Anaya’s Bless Me Última, Tomás Rivera’s …Y No Se Lo Tragó La Tierra, and Rolando Hinojosa’s Los Amigos de Becky. The paper proposes an additional application of “hybridity”, key concept of Cultural and Postcolonial Studies, in the cultural context of the Chicano identity. The fi nal aim is to prove, by means of the texts, the interaction between the individual and his/her context via their “hybridity” as a resulting phenomenon of a cultural multiplicity. Este trabajo presenta un análisis de tres novelas de autores chicanos considerados canónicos, en el que se discute el concepto de ‘hibridación’ y su interacción con el individuo: Bless Me Última, de Rudolfo Anaya, …Y No Se Lo Tragó La Tierra, de Tomás Rivera y Los Amigos de Becky, de Rolando Hinojosa. Se propone una aplicación adicional de “hibridación”, concepto clave de los Estudios Culturales y Postcoloniales, en el contexto cultural de la identidad chicana. El objetivo es demostrar la interacción entre el individuo y su contexto a través de la “hibridación” como fenómeno de multiplicidad cultural

On the Xiao conjecture for plane curves

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