3,446 research outputs found
Deformation classification of real non-singular cubic threefolds with a marked line
We prove that the space of pairs formed by a real non-singular cubic hypersurface with a real line has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface formed by real lines on . For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of characterizes completely the component
Prym varieties of genus four curves
Double covers of a generic genus four curve C are in bijection with Cayley
cubics containing the canonical model of C. The Prym variety associated to a
double cover is a quadratic twist of the Jacobian of a genus three curve X. The
curve X can be obtained by intersecting the dual of the corresponding Cayley
cubic with the dual of the quadric containing C. We take this construction to
its limit, studying all smooth degenerations and proving that the construction,
with appropriate modifications, extends to the complement of a specific divisor
in moduli. We work over an arbitrary field of characteristic different from two
in order to facilitate arithmetic applications.Comment: 30 pages; Some expository changes; removed erroneous (old) Thm 4.11
and changed (old) Thm 4.23 into (new) Thm 4.1
Arithmetic aspects of the Burkhardt quartic threefold
We show that the Burkhardt quartic threefold is rational over any field of
characteristic distinct from 3. We compute its zeta function over finite
fields. We realize one of its moduli interpretations explicitly by determining
a model for the universal genus 2 curve over it, as a double cover of the
projective line. We show that the j-planes in the Burkhardt quartic mark the
order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse
pencil on a j-plane gives rise to the universal curve as a discriminant of a
cubic genus one cover.Comment: 22 pages. Amended references to more properly credit the classical
literature (Coble in particular
On singular Luroth quartics
Plane quartics containing the ten vertices of a complete pentalateral and
limits of them are called L\"uroth quartics. The locus of singular L\"uroth
quartics has two irreducible components, both of codimension two in .
We compute the degree of them and we discuss the consequences of this
computation on the explicit form of the L\"uroth invariant. One important tool
are the Cremona hexahedral equations of the cubic surface. We also compute the
class in of the closure of the locus of nonsingular L\"uroth
quartics.Comment: Enlarged version. Computation of the class of the locus of Luroth
quartics in the moduli space adde
- …