Related papers: Orbit parametrizations of theta characteristics on…
Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the…
In the present paper we study the geometry of plane quartics with large automorphism groups. We show results devoted to smooth plane quartics that are invariant under the action of the elementary abelian group of type $[2,2,2]$, and we…
Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…
We determine the automorphism group for a large class of affine quadric hypersurfaces over a field, viewed as affine algebraic varieties. In particular, we find that the group of real polynomial automorphisms of the n-sphere is just the…
For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…
The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and…
We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in…
We study closed smooth convex plane curves $\Gamma$ enjoying the following property: a pair of points $x,y$ can traverse $\Gamma$ so that the distances between $x$ and $y$ along the curve and in the ambient plane do not change; such curves…
We study main bifurcations of multidimensional diffeomorphisms having a non-transversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a…
We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…
We give two characterizations of hyperquadrics: one as non-degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as $LQEL$-manifolds with large secant defects.
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating…
We study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it.
Merkurjev's theorem--the statement that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms--is in general false when the base is no longer a field. The work of Parimala, Scharlau, and Sridharan proves…
Let S be an nXn real symmetric matrix with spectral decomposition S=Q^T Lambda Q, where Q is an orthogonal matrix and Lambda is diagonal with simple spectrum {lambda_1,..., lambda_n}. Also let O_S e R_S be the orbits by conjugation of S by,…
We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that…
We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.
We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth…
Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…