Anthony Iarrobino

Compressed algebras: Artin algebras having given socle degrees and maximal length

J. Emsalem and the author showed in [18] that a general polynomialf of degree j in the ring SP= k[yl,...,yr] has (jr+-rl 1) linearly independent partial derivates of order i, for i = 0,1, . . ., t = [ j/2]. Here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible, given the room available. Using this result, we construct and describe the varieties G( E) and Z( E) parametrizing the graded and nongraded compressed algebra quotients A = R/I of the power series ring R = k[[xl sXr]] having given socle type E. These algebras are Artin algebras having maximal length dimk A possible, given the embedding degree r and given the socle-type sequence E = ( els . . . s eS), where ei is the number of generators of the dual module A of A, having degree i. The variety Z(E) is locally closed, irreducible, and is a bundle over G(E), fibred by affine spaces AN whose dimension is known. We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable-have no deformation to (k + + k)-for dimension reasons. For some choices of the sequence E, D. Buchsbaum, D. Eisenbud and the author have shown that the graded compressed algebras of socle-type E have almost linear minimal resolutions over R, with ranks and degrees determined by E. Other examples have given type e = dimk (socle A) and are defined by an ideal I with certain given numbers of generators in R =

Associated graded algebra of a Gorenstein Artin algebra

Gorenstein Artin algebras and duality The intersection of two plane curves Extremal decompositions Components of the Hilbert scheme strata What decompositions

A nonunimodal graded gorenstein artin algebra in codimension five

D

Ancestor ideals of vector spaces of forms, and level algebras

and subquotients

The Family G T of Graded Artinian Quotients of k[x, y] of Given Hilbert Function

Q(a)

The Hilbert function of a Cohen-Macaulay local algebra: Extremal gorenstein algebras

can occur? Relatively compressed Artin algebras Bibliography List of theorems, definitions, and examples Index.

Strata of rational space curves

A graded standard Gorenstein Artin algebra quotient of the polynomial ring R over k can be viewed as the algebra Af of partial differential operators of all degrees on a form F. The algebra A is unimodal if the Hilbert function has a single local maximum. We use the theory of compressed algebras to construct forms F in five or more variables whose Gorenstein algebras Af are not unimodal.

High-order vanishing ideals at n+3 points of Pn

Let R=k[x1,…,xr] denote the polynomial ring in r variables over a field k, with maximal ideal M=(x1,…,xr), and let V⊂Rj denote a vector subspace of the space Rj of degree-j homogeneous elements of R. We study three related algebras determined by V. The first is the ancestor algebra Anc(V)=R/V whose defining ancestor ideal V is the largest graded ideal of R such that V∩Mj=(V), the ideal generated by V. The second is the level algebra LA(V)=R/L(V) whose defining ideal L(V), is the largest graded ideal of R such that the degree-j component L(V)∩Rj is V; and third is the algebra R/(V). We have that L(V)=V+Mj+1. When r=2 we determine the possible Hilbert functions H for each of these algebras, and as well the dimension of each Hilbert function stratum. We characterize the graded Betti numbers of these algebras in terms of certain partitions depending only on H, and give the codimension of each stratum in terms of invariants of the partitions. We show that when r=2 and k is algebraically closed the Hilbert function strata for each of the three algebras attached to V satisfy a frontier property that the closure of a stratum is the union of more special strata. In each case the family G(H) of all graded ideals of the given Hilbert function is a natural desingularization of this closure. We then solve a refinement of the simultaneous Waring problem for sets of degree-j binary forms. Key tools throughout include properties of an invariant τ(V), the number of generators of V⊂k[x1,x2], and previous results concerning the projective variety G(H) in [Mem. Amer. Math. Soc., Vol. 10 (188), 1977].

The Hilbert Function of a Gorenstein Artin Algebra

Abstract Let R = k[x, y] be the polynomial ring over an algebraically closed field k. Let Tbe a sequence of nonnegative integers that occurs as the Hilbert function of a length-nArtinian quotient of R. The nonsingular projective variety G T parametrizes all graded ideals Iof R = k[x, y] for which the Hilbert function H(R/I) = T(see Iarrobino, A. (1977). Punctual Hilbert Schemes. Mem. Amer. Math. Soc. Vol. 10, #188, Providence: American Mathematical Society). We show that G T is birational to a certain product SGrass(T) of small Grassmann varieties (Proposition 3.15), and that over k = ℂ the birational map induces an additive ℤ-isomorphism τ : H*(G T ) → H*(SGrass(T)) of homology groups (Theorem 3.29). The map τ is not usually an isomorphism of rings. We determine the ring H*(G T ) when where G T ⊂ ℙυ×ℙ j (Theorem 4.5). In this case G T is a desingularisation of the υ-secant bundle Sec(υ, j) of the degree jrational normal curve. We use this ring H*(G T ) to determine the number of ideals satisfying an intersection of ramification conditions at different points (Example 4.6). We also determine the classes in H*(G T ) of the pullback of the singular locus of Sec(υ, j) and of the pullbacks of the higher singular loci (Theorem 4.12). Let Ebe a monomial ideal of R, satisfying H(R/E) = T, where ∣ T ∣ = n: it corresponds to a partition P(E) of nhaving diagonal lengths T. A main tool is that the family of graded ideals having initial monomials Eis a cell 𝕍(E). We connect these cells to ramification conditions, using the Wronskian determinant, and to a “hook code” for P(E). Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Punctual Hilbert schemes

algebras, studied their resolutions, and extended Theorem 1 to the case A Cohen-Macaulay graded [F-L]. We extend the inequality to C.M. local algebras A that need not be graded. Here the associated graded algebra A* need not be Cohen-Macaulay; even if A* is CM., its socle type may be dif- ferent from that of A. Assume that I is an ideal defining a Gorenstein quotient A of