arXiv subjects
Publications and source records attributed to H. Lange.
In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems introduced by Hernández Ruiperez and Tejero Prieto.
Let $G$ denote a finite group and $π: Z \to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\mathbb{Q}[H \backslash G/H]$ on the Jacobian of the curve $X = Z/H$. To each rational irreducible representation $\mathcal{W}$ of $G$ we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve $X$ and thus an abelian subvariety $P$ of the Jacobian $JX$. We give sufficient conditions on $\mathcal{W}$, $H$, and the action of $G$ on $Z$, which imply $P$ to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way.
Given Prym-Tyurin varieties of exponent $q$ with respect to a finite group $G$, a subgroup $H$ and a set of rational irreducible representations of $G$ satisfying some additional properties, we construct a Prym-Tyurin variety of exponent $[G:H]q$ in a natural way. We study an example of this result, starting from the dihedral group $\mathbf{D}_p$ for any odd prime $p$. This generalizes the construction of arXiv:math/0412103v2[math.AG] for $p=3$. Finally, we compute the isogeny decomposition of the Jacobian of the curve underlying the above mentioned example.
Let $X_1, ..., X_m$ denote smooth projective curves of genus $g_i \geq 2$ over an algebraically closed field of characteristic 0 and let $n$ denote any integer at least equal to $1+\max_{i=1}^m g_i$. We show that the product $JX_1 \times ... \times JX_m$ of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent $n^{m-1}$. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.
The Prym map of type (g,n,r) associates to every cyclic covering of degree n of a curve of genus g, ramified at a reduced divisor of degree r, the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.
We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tjurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.
In this paper we continue the investigation of coherent systems of type $(n,d,k)$ on the projective line which are stable with respect to some value of a parameter $α$. We consider the case $k=1$ and study the variation of the moduli spaces with $α$. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.
We construct the coarse moduli space $\cM_{qc}(σ)$ of quadratic line complexes with a fixed Segre symbol $σ$ as well as the moduli space $\cM_{ss}(σ)$ of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism $π: \cM_{qc}(σ) \ra \cM_{ss}(σ)$. Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.
Let $G$ be a finite group, $Λ$ an absolutely irreducible $\Z[G]$-module and $w$ a weight of $Λ$. To any Galois covering with group $G$ we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym-Tyurin varieties.
In this paper we continue the investigation of coherent systems of type (n,d,k) on the projective line which are stable with respect to some value of the parameter α. We work mainly with k<n and obtain existence results for arbitrary k in certain cases, together with complete results for k=3. Our methods involve the use of the "flips" which occur at critical values of the parameter.
To any pair of coverings $f_i: X \ra X_i, i = 1,2$ of smooth projective curves one can associate an abelian subvariety of the Jacobian $J_X$, the Prym variety $P(f_1,f_2)$ of the pair $(f_1,f_2)$. In some cases we can compute the type of the restriction of the canonical principal polarization of $JX$. We obtain 2 families of Prym-Tyurin varieties of exponent 6.
In this paper we consider coherent systems $(E,V)$ on an elliptic curve which are stable with respect to some value of a parameter $α$. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimenson. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with $α$.
We investigate a family of correspondences associated to étale coverings of degree 3 of hyperelliptic curves. They lead to Prym-Tyurin varieties of exponent 3. We identify these varieties and derive some consequences.
Let $M$ denote the moduli space of stable vector bundles of rank $n$ and fixed determinant of degree coprime to $n$ on a non-singular projective curve $X$ of genus $g \geq 2$. Denote by $\cU$ a universal bundle on $X \times M$. We show that, for $x,y \in X, x \neq y$, the restrictions $\cU|\{x\} \times M$ and $\cU|\{y\} \times M$ are stable and non-isomorphic when considered as bundles on $X$.
In this paper we begin the classification of coherent systems $(E,V)$ on the projective line which are stable with respect to some value of a parameter $α$. In particular we show that the moduli spaces, if non-empty, are always smooth and irreducible of the expected dimension. We obtain necessary conditions for non-emptiness and, when $\dim V=1$ or 2, we determine these conditions precisely. We also obtain partial results in some other cases.
Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of JX up to isogeny. The most prominent example of such a situation is the group G of two elements. Let X --> Y denote the corresponding quotient map. Then JX is isogenous to the product of JY with the Prym variety of X/Y. In this paper some general results on group actions on abelian varieties are given and applied to deduce a decomposition of the jacobian JX for arbitrary group actions. Several examples are given.
In this paper we obtain bounds on $h^0(E)$ where $E$ is a semistable bundle of rank 3 over a smooth irreducible projective curve $X$ of genus $g \geq 2$ defined over an algebraically closed field of characteristic 0. These bounds are expressed in terms of the degrees of stability $s_1(E), s_2(E)$. We show also that in some cases the bounds are best possible. These results extend recent work of J. Cilleruelo and I. Sols for bundles of rank 2.