Each partition λ = (λ1, λ2, . . . , λn) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs amo...

KrulΓs principal ideal theorm [Krull] states that q elements in the maximal ideal of a local noetherian ring generate an ideal whose minimal components are all of height at most q. Writing R for the ring, we may consider the q elements, x19 , xq say, as coordinates of an element xeR. It is an easy observation that every homomorphism R —> R carries x to an element of the ideal generated by xi9 ,...

We construct harmonic diffeomorphisms from the complex plane C onto any Hadamard surface M whose curvature is bounded above by a negative constant. For that, we prove a JenkinsSerrin type theorem for minimal graphs in M × R over domains of M bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in M...

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger. Let R = k[x1, x2, . . . , xr] be a polynomial ring over a field. We will say that an ideal I ⊆ R has the saturated chain p...

In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated lcm-lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose lcm-lattice is P , and we give a characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atom...

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids M such that its corresponding toric ideal IM is a complete intersection. Secondly, we handle the problem of detecting minors of a matroidM from a minimal set of binomial generators of IM. In particular, given a minimal set of binomial generators of IM we provide a neces...

We present a constructive description of minimal reductions with a given reduction number. This description has interesting consequences on the minimal reduction number, the big reduction number, and the core of an ideal. In particular, it helps solve a conjecture of Vasconcelos on the relationship between reduction numbers and initial ideals.

We show that the minimal cardinality of a dense subset of the measure algebra is the same as the minimal cardinality of a base of the ideal of Lebesgue measure zero subsets of the real line. 0. Introduction. Let (F, <) be a given partial ordering. A subset D ç P is called dense if for any p g F there exists d g D such that d < p. A subset D is called upward dense if D is dense in (F, > ). Let A...

We construct complete embedded minimal surfaces in H × R. The first one is a finite total curvature surface which is conformal to S \ {p1, ..., pk}, k ≥ 2; the second one is a 1-parameter family of singly-periodic minimal surfaces which is asymptotic to a horizontal plane and a vertical plane; the third one is a 2-parameter family of minimal surfaces which have a fundamental piece of finite tot...