arXiv subjects

H. Lange

Publications and source records attributed to H. Lange.

43 records · Page 3Linked to original sources

arXiv2002

Maximal Subbundles and Gromov-Witten Invariants

Let $C$ be a nonsingular irreducible projective curve of genus $g\ge2$ defined over the complex numbers. Suppose that $1\le n'\le n-1$ and $n'd-nd'=n'(n-n')(g-1)$. It is known that, for the general vector bundle $E$ of rank $n$ and degree $d$, the maximal degree of a subbundle of $E$ of rank $n'$ is $d'$ and that there are finitely many such subbundles. We obtain a formula for the number of these maximal subbundles when $(n',d')=1$. For $g=2$, $n'=2$, we evaluate this formula explicitly. The numbers computed here are Gromov-Witten invariants in the sense of a recent paper of Ch. Okonek and A. Teleman (to appear in Commun. Math. Phys.) and our results answer a question raised in that paper. In this revised version some references are added.

math.AG↗

arXiv2002

An Isomorphism Between Moduli Spaces of Abelian Varieties

In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main consequence we obtain an involution on the set of moduli spaces of polarized abelian varieties of dimension $g$. In particular, the moduli spaces $\mathcal A_{(d_1,...,d_g)}$ and $\mathcal A_{(d_1,\frac{d_1d_g}{d_{g-1}},...,\frac{d_1d_g}{d_2},d_g)}$ are canonically isomorphic.

math.AG↗

arXiv2001

Severi varieties and branch curves of abelian surfaces of type (1,3)

Let (A,L) be a principally polarized abelian surface of type (1,3). The linear system |L| defines a 6:1 covering of A onto P2, branched along a curve B of degree 18 in P2. The main result of the paper is that for general (A,L) the curve B irreducible, admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as two nodes, 72 flexes and 36 bitangents. The main idea of the proof is to use the fact that for a general (A,L) of type (1,3) the closure of the Severi variety V in |L| is dual to the curve B in the sense of projective geometry. We investigate V and B via degeneration to a special abelian surface.

math.AG↗

arXiv1998

Theoretical modelling of Langmuir monolayers

We study coarse grained, continuum models for Langmuir monolayers by self consistent field theory and by Monte Carlo simulations. Amphiphilic molecules are represented by stiff chains of monomers with one end grafted to a planar surface. In particular, we discuss the origin of successive fluid-fluid transitions, the possible origin of tilt order and the factors which determine the direction of tilt.

cond-mat.soft↗

arXiv1997

A stratification of the moduli space of vector bundles on curves

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k°E-r\max°F.$$ where the maximum is taken over all subbundles $F$ of rank $k$ of $E$. The ${s}_k$ gives a stratification of the moduli space ${\cal M}(r,d)$ of stable vector bundles of rank $r$ and degree on $d$ on $C$ into locally closed subsets ${\calM}(r,d,k,s)$ according to the value of $s$ and $k$. There is a component ${\cal M}^0(r,d,k,s)$ of ${\cal M}(r,d,k,s)$ distinguish by the fact that a general $E\in {\cal M}^0(r,d,k,s)$ admits a stable subbundle $F$ such that $E/F$ is also stable. We prove: {\it For $g\ge \frac{r+1}{2}$ and $0<s\leq k(r-k)(g-1) +(r+1)$, $s\equiv kd \mod r,$ ${\cal M}^0(r,d,k,s)$ is non-empty,and its component ${\cal M}^0(r,d,k,s)$ is of dimension} $$\dim {\cal M}^0(r,d,k,s)=\left\{\begin{array}{lcl} (r^2+k^2-rk)(g-1)+s-1& &s<k(r-k)(g-1) &{\rm if}& r^2(g-1)+1& & s\ge k(r-k)(g-1)\end{array}\right.$$

alg-geom↗

arXiv1996

Influence of the Head Group Size on the Direction of Tilt in Langmuir Monolayers

A model of rods with heads of variable size, which are confined to a planar surface, is used to study the influence of the head group size on tilted phases in Langmuir monolayers. Simple free energy considerations as well as exact zero temperature calculations indicate that molecules with small head groups tilt towards next nearest neighbors, and molecules with larger head groups towards nearest neighbors. This provides a possible explanation for recent experimental results, and for details of the generic phase diagram for fatty acid monolayers.

physics.chem-ph↗

arXiv1995

A Vector Bundle of Rank 2 on P1 x P3

The paper studies a rank 2 vector bundle on P1 x P3. Similarly to the Horrocks - Mumford bundle on P4 this vector bundle encodes a lot of geometric information. It is defined via the Serre construction by an abelian surface in P1 x P3. The bundle is stable with respect to O(1,1), its jumping lines are determined. Morover it is shown that it is closely related to a double structue on P1 x P1 embedded in P1 x P3 as well as to a certain pencil of quadrics of degree 4 in P3.

alg-geom↗