19 research outputs found

    Extensors and the Hilbert scheme

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    The Hilbert scheme Hilbp(t)n\mathbf{Hilb}_{p(t)}^{n} parametrizes closed subschemes and families of closed subschemes in the projective space Pn\mathbb{P}^n with a fixed Hilbert polynomial p(t)p(t). 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 kk of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian Grp(r)N(r)\mathbf{Gr}_{p(r)}^{N(r)}, where N(r)=h0(OPn(r))N(r)= h^0 (\mathcal{O}_{\mathbb{P}^n}(r)). Moreover, we exhibit explicit equations defining it in the Pl\"ucker coordinates of the Pl\"ucker embedding of Grp(r)N(r)\mathbf{Gr}_{p(r)}^{N(r)}. 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 degp(t)+2\text{deg} p(t)+2, 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

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    Let GG 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 GG with a given infinitesimal character and a given bound in the complex associated variety. When GG is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of GG attached to Oˇ\check{\mathcal O}, in the sense of Barbasch and Vogan. Here Oˇ\check{\mathcal O} is a nilpotent adjoint orbit in the Langlands dual of GG (or the metaplectic dual of GG when GG is a real metaplectic group). We give a precise count for the number of special unipotent representations of GG attached to Oˇ\check{\mathcal O}. We also reduce the problem of constructing special unipotent representations attached to Oˇ\check{\mathcal O} to the case when Oˇ\check{\mathcal O} 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 nn achieve nearly the optimal O(nα1/2)\mathcal{O}(n^{-\alpha-1/2}) error independently of the dimension

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    We analyze a new random algorithm for numerical integration of dd-variate functions over [0,1]d[0,1]^d from a weighted Sobolev space with dominating mixed smoothness α0\alpha\ge 0 and product weights 1γ1γ2>01\ge\gamma_1\ge\gamma_2\ge\cdots>0, where the functions are continuous and periodic when α>1/2\alpha>1/2. The algorithm is based on rank-11 lattice rules with a random number of points~nn. For the case α>1/2\alpha>1/2, we prove that the algorithm achieves almost the optimal order of convergence of O(nα1/2)\mathcal{O}(n^{-\alpha-1/2}), where the implied constant is independent of the dimension~dd if the weights satisfy j=1γj1/α<\sum_{j=1}^\infty \gamma_j^{1/\alpha}<\infty. The same rate of convergence holds for the more general case α>0\alpha>0 by adding a random shift to the lattice rule with random nn. This shows, in particular, that the exponent of strong tractability in the randomized setting equals 1/(α+1/2)1/(\alpha+1/2), 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-11 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
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