38 research outputs found
On first order Congruences of Lines in with irreducible fundamental Surface
In this article we study congruences of lines in , and in
particular of order one. After giving general results, we obtain a complete
classification in the case of in which the fundamental surface
is in fact a variety-i.e. it is integral-and the congruence is the
irreducible set of the trisecant lines of .Comment: 18 pages, AMS-LaTeX; submitte
Threefolds of P^5 with one apparent quadruple point
In this article we classify all the smooth threefolds of P^5 with an apparent
quadruple point provided that the family of its 4-secant lines is an
irreducible (first order) congruence. This is sufficient to conclude the
classification of all the smooth codimension two varieties of P^n with one
apparent (n-1)-point and with irreducible family of (n-1)-secant lines.Comment: AMS-LaTeX2e, 14 page
Gonality, apolarity and hypercubics
We show that any Fermat hypercubic is apolar to a trigonal curve, and vice
versa. We show also that the Waring number of the polar hypercubic associated
to a tetragonal curve of genus is at most , and
for a large class of them is at most .Comment: 9 pages, to appear in the Bulletin of the London Mathematical Societ
On higher Gauss maps
We prove that the general fibre of the -th Gauss map has dimension if
and only if at the general point the -th fundamental form consists of
cones with vertex a fixed , extending a known theorem for the
usual Gauss map. We prove this via a recursive formula for expressing higher
fundamental forms. We also show some consequences of these results.Comment: 12 pages, AMS-LaTeX; to appear in the Journal of Pure and Applied
Algebr
Fano congruences of index and alternating -forms
We study congruences of lines defined by a sufficiently general
choice of an alternating 3-form in dimensions, as Fano manifolds
of index and dimension . These congruences include the
-variety for and the variety of reductions of projected
for .
We compute the degree of as the -th Fine number and study the
Hilbert scheme of these congruences proving that the choice of
bijectively corresponds to except when . The fundamental locus
of the congruence is also studied together with its singular locus: these
varieties include the Coble cubic for and the Peskine variety for .
The residual congruence of with respect to a general linear
congruence containing is analysed in terms of the quadrics
containing the linear span of . We prove that is Cohen-Macaulay
but non-Gorenstein in codimension . We also examine the fundamental locus
of of which we determine the singularities and the irreducible
components.Comment: 46 pages, 2 tables. AMS-LaTeX. Minor changes. To appear in the
Annales de l'Institut Fourie
On the hypersurfaces contained in their Hessian
This article presents the theory of focal locus applied to the hyper-
surfaces in the projective space which are (finitely) covered by linear spaces
and such that the tangent space is constant along these spaces