Sur le schéma de Hilbert des variétés de codimension $2$ dans $\mathbf{P}^e$ à cône de Cohen-Macaulay

Abstract

Sur le schéma de Hilbert des variétés de codimension 2 dans P e à cône de Cohen-Macaulay Annales scientifiques de l'É.N.S. 4 e série, tome 8, n o 4 (1975), p. 423-431 <http://www.numdam.org/item?id=ASENS_1975_4_8_4_423_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1975, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Dimension of families of determinantal schemes

“…So, from now on, we assume (c − 1)a t+c−2 ≥ and we will proceed by induction on c by successively deleting columns from the right side, i.e., of the largest degree. For c = 2 the result was proved in [12] if n ≥ 1 and in [19] for any n ≥ 0. So, we will assume c ≥ 3.…”

“…In turns out that the upper bound of dim W (b; a) given in Theorem 3.5 is sharp in a number of instances. More precisely, if 2 ≤ c ≤ 3, this is known ( [19], [12]), for 4 ≤ c ≤ 5 it is a consequence of the main theorem of this section (see Corollaries 4.10 and 4.14), while for c ≥ 6 we get the expected dimension formula for W (b; a) under more restrictive assumptions (see Corollary 4.15). In Section 5, we study when the closure of W (b; a) is an irreducible component of Hilb p (P n+c ) and when Hilb p (P n+c ) is generically smooth along W (b; a), and other cases of unobstructedness.…”

“…good) determinantal schemes. The first important contribution to this problem is due to G. Ellinsgrud [12]; in 1975, he proved that every arithmetically Cohen-Macaulay, codimension 2 closed subscheme X of P n+2 is unobstructed (i.e., the corresponding point in the Hilbert scheme Hilb p (P n+2 ) is smooth) provided n ≥ 1 and he also computed the dimension of the Hilbert scheme at X. Recall also that the homogeneous ideal of an arithmetically Cohen-Macaulay, codimension 2 closed subscheme X of P n+2 is given by the maximal minors of a t × (t + 1) homogeneous matrix, the Hilbert-Burch matrix.…”

Ideals generated by submaximal minors

“…(1) determining the dimension of W r p,q (b; a) in terms of a j and b i , Ellingsrud [1975], who proved that every arithmetically Cohen-Macaulay, closed subscheme X of codimension 2 of ‫ސ‬ n is unobstructed (that is, the corresponding point in the Hilbert scheme Hilb p(x) ‫ސ(‬ n ) is smooth) provided n ≥ 3. He also computed the dimension of the Hilbert scheme at (X ).…”

On Characteristic Classes of Determinantal Cremona Transformations

“…The family of base schemes of determinantal maps, not necessarily birational, may be identified as an open and connected subset of the Hilbert scheme of the 2-codimensional arithmetically Cohen-Macaulay subschemes of P n (see [6]). …”