19 research outputs found
Extensors and the Hilbert scheme
The Hilbert scheme parametrizes closed subschemes
and families of closed subschemes in the projective space with a
fixed Hilbert polynomial . It is classically realized as a closed
subscheme of a Grassmannian or a product of Grassmannians. In this paper we
consider schemes over a field of characteristic zero and we present a new
proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian
, where .
Moreover, we exhibit explicit equations defining it in the Pl\"ucker
coordinates of the Pl\"ucker embedding of . Our
proof of existence does not need some of the classical tools used in previous
proofs, as flattening stratifications and Gotzmann's Persistence Theorem. The
degree of our equations is , lower than the degree of the
equations given by Iarrobino and Kleiman in 1999 and also lower (except for the
case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels
in 2004 after Bayer's conjecture in 1982. The novelty of our approach mainly
relies on the deeper attention to the intrinsic symmetries of the Hilbert
scheme and on some results about Grassmannian based on the notion of extensors.Comment: Added equations of the Hilbert schemes of 2 points in the plane,
3-space, 4-space and of 3 points in the plane (a Macaulay2 file with the
complete computation is available at
http://tinyurl.com/EquationsHilbPoints-m2). Final version. To appear on
Annali della Scuola Normale Superiore di Pisa - Classe di Scienz
Special unipotent representations of real classical groups: counting and reduction to good parity
Let be a real reductive group in Harish-Chandra's class. We derive some
consequences of theory of coherent continuation representations to the counting
of irreducible representations of with a given infinitesimal character and
a given bound in the complex associated variety. When is a real classical
group (including the real metaplectic group), we investigate the set of special
unipotent representations of attached to , in the sense
of Barbasch and Vogan. Here is a nilpotent adjoint orbit
in the Langlands dual of (or the metaplectic dual of when is a real
metaplectic group). We give a precise count for the number of special unipotent
representations of attached to . We also reduce the
problem of constructing special unipotent representations attached to
to the case when has good parity. The
paper is the first in a series of two papers on the classification of special
unipotent representations of real classical groups.Comment: 77 page
Lattice rules with random achieve nearly the optimal error independently of the dimension
We analyze a new random algorithm for numerical integration of -variate
functions over from a weighted Sobolev space with dominating mixed
smoothness and product weights
, where the functions are continuous and
periodic when . The algorithm is based on rank- lattice rules
with a random number of points~. For the case , we prove that
the algorithm achieves almost the optimal order of convergence of
, where the implied constant is independent of
the dimension~ if the weights satisfy . The same rate of convergence holds for the more
general case by adding a random shift to the lattice rule with
random . This shows, in particular, that the exponent of strong tractability
in the randomized setting equals , if the weights decay fast
enough. We obtain a lower bound to indicate that our results are essentially
optimal. This paper is a significant advancement over previous related works
with respect to the potential for implementation and the independence of error
bounds on the problem dimension. Other known algorithms which achieve the
optimal error bounds, such as those based on Frolov's method, are very
difficult to implement especially in high dimensions. Here we adapt a
lesser-known randomization technique introduced by Bakhvalov in 1961. This
algorithm is based on rank- lattice rules which are very easy to implement
given the integer generating vectors. A simple probabilistic approach can be
used to obtain suitable generating vectors.Comment: 17 page