3,830 research outputs found
On the discriminant scheme of homogeneous polynomials
arXiv reference : arXiv:1210.4697International audienceIn this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of n-1 homogeneous polynomials in n variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring k, typically a domain, if 2 is not equal to 0 in k. The case where 2 is equal to 0 in k is also analyzed in detail
The scheme of liftings and applications
We study the locus of the liftings of a homogeneous ideal in a polynomial
ring over any field. We prove that this locus can be endowed with a structure
of scheme by applying the constructive methods of Gr\"obner
bases, for any given term order. Indeed, this structure does not depend on the
term order, since it can be defined as the scheme representing the functor of
liftings of . We also provide an explicit isomorphism between the schemes
corresponding to two different term orders.
Our approach allows to embed in a Hilbert scheme as a locally
closed subscheme, and, over an infinite field, leads to find interesting
topological properties, as for instance that is connected and
that its locus of radical liftings is open. Moreover, we show that every ideal
defining an arithmetically Cohen-Macaulay scheme of codimension two has a
radical lifting, giving in particular an answer to an open question posed by L.
G. Roberts in 1989.Comment: the presentation of the results has been improved, new section
(Section 6 of this version) concerning the torus action on the scheme of
liftings, more detailed proofs in Section 7 of this version (Section 6 in the
previous version), new example added (Example 8.5 of this version
The Cayley-Oguiso automorphism of positive entropy on a K3 surface
Recently Oguiso showed the existence of K3 surfaces that admit a fixed point
free automorphism of positive entropy. The K3 surfaces used by Oguiso have a
particular rank two Picard lattice. We show, using results of Beauville, that
these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley
constructed an automorphism of such determinantal surfaces. We show that
Cayley's automorphism coincides with Oguiso's free automorphism. We also
exhibit an explicit example of a determinantal quartic whose Picard lattice has
exactly rank two and for which we thus have an explicit description of the
automorphism.Comment: 22 pages, 1 figure. We added several improvements, as well as a
figure. A smaller pdf file with the figure in lower resolution (faster on
most viewers) is available by downloading the source under "Other formats
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