50 research outputs found
On the Alexander-Hirschowitz Theorem
The Alexander-Hirschowitz theorem says that a general collection of
double points in imposes independent conditions on homogeneous
polynomials of degree with a well known list of exceptions. Alexander and
Hirschowitz completed its proof in 1995, solving a long standing classical
problem, connected with the Waring problem for polynomials. We expose a
self-contained proof based mainly on previous works by Terracini, Hirschowitz,
Alexander and Chandler, with a few simplifications. We claim originality only
in the case , where our proof is shorter. We end with an account of the
history of the work on this problem.Comment: 29 pages, the proof in the case of cubics has been simplified, three
references added, to appear in J. Pure Appl. Algebr
On the hypersurface of Luroth quartics
The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface