On binomial complete intersections

The non-Lefschetz locus

Syzygies in Hilbert schemes of complete intersections

This paper proves that the Castelnuovo-Mumford regularities of the product and sum of two monomial complete intersection ideals are at most the sum of the regularities of the two ideals, and provides examples showing that these inequalities do not hold for general complete intersections.

Let A A be a graded complete intersection over a field and B B the monomial complete intersection with the generators of the same degrees as A A . The EGH conjecture says that if I I is a graded ideal in A A , then there should be an ideal J J in B B such that B / J B/J and A / I A/I have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.

Denote by [Formula: see text] a polynomial ring over a field, and let [Formula: see text] be a monomial ideal of [Formula: see text]. If [Formula: see text], we prove that the multiplicity of [Formula: see text] is given by [Formula: see text] On the other hand, if [Formula: see text] is a complete intersection, and [Formula: see text] is an almost complete intersection, we show that [Formula: see text] We also introduce a new class of ideals that extends the family of monomial complete intersections and that of codimension 1 ideals, and give an explicit formula for their multiplicity.

The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+\langle f \rangle$ for an explicit polynomial $f$ constructed from the fibers of the Veronese grading map.

The primary components of positive critical binomial ideals

Let $V$ be an infinite-dimensional complex Banach space and $X \subset {\mathbf {P}}(V)$ a closed analytic subset with finite codimension. We give a condition on $X$ which implies that $X$ is a complete intersection. We conjecture that the result should be true for more general topological vector spaces.

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char K∤∣G∣. We show that if the algebra of invariants K(V)G of the symmetric algebra of V is a complete intersection then K(V)H is also a complete intersection for all subgroups H of G such that H={σ ε Gv σ(v)=v for all v ε VH}.

Separating invariants of finite groups

Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X, Y, Z]/(X d , Y d , Z d ) in terms of d and p. This answers a question of Migliore, Miró-Roig and Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X d , Y d , Z d ) on $${{\mathbb {P}}^2}$$ satisfies the Grauert-Mülich theorem. As a corollary we obtain that for p = 2 the algebra A has the weak Lefschetz property if and only if $${d=\lfloor\frac{2^t+1}{3}\rfloor}$$ for some positive integer t. This was recently conjectured by Li and Zanello.

We compute the Stanley depth for a particular but important case of the quotient of complete intersection monomial ideals. Also, in the general case, we give sharp bounds for the Stanley depth of a quotient of complete intersection monomial ideals. In particular, we prove the Stanley conjecture for quotients of complete intersection monomial ideals.

Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections

We determine the rings of invariants $S^G$ where $S$ is the symmetric algebra on the dual of a vector space $V$ over ${\mathbb F}_2$ and $G$ is the orthogonal group preserving a non-singular quadratic form on $V$. The invariant ring is shown to have a presentation in which the difference between the number of generators and the number of relations is equal to the minimum possibility, namely $\dim V$, and it is shown to be a complete intersection. In particular, the rings of invariants computed here are all Gorenstein and hence Cohen-Macaulay

Modular invariants of a vector and a covector: A proof of a conjecture of Bonnafé and Kemper