arXiv subjects
Publications and source records attributed to Ignacio Ojeda.
Let $S_1$ and $S_2$ be two affine semigroups and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are then related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.
In this paper a new classification of monomial curves in $\mathbb{A}^4(\mathbbmss{k})$ is given. Our classification relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient condition for the definining ideal of a monomial curve in $\mathbb{A}^4(\mathbbmss{k})$ to have a unique minimal system of binomial generators. Furthermore, some other interesting results on more general classes of binomial ideals with unique minimal system of binomial generators are obtained.
Let $\mathbb{N} \mathcal{A}$ be the monoid generated by $\mathcal{A} = {\mathbf{a}_1, ..., \mathbf{a}_n} \subseteq \mathbb{Z}^d.$ We introduce the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ as the smallest $N \in \mathbb N$ with the following property: for each $\mathbf{a} \in \mathbb{N} \mathcal{A}$ and any two factorizations $\mathbf{u}, \mathbf{v}$ of $\mathbf{a}$, there exists factorizations $\mathbf{u} = \mathbf{w}_1, ..., \mathbf{w}_t = \mathbf{v} $ of $\mathbf{a}$ such that, for every $k, \mathrm{d}(\mathbf{w}_k, \mathbf{w}_{k+1}) \leq N,$ where $\mathrm{d}$ is the usual distance between factorizations, and the length of $\mathbf{w}_k, |\mathbf{w}_k|,$ is less than or equal to $\max{|\mathbf{u}|, |\mathbf{v}|}.$ We prove that the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the $ω$-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.
We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups. As an example, we particularize our description to numerical semigroups.
In this paper, we prove that every binomial ideal in a polynomial ring over an algebraically closed field of characteristic zero admits a canonical primary decomposition into binomial ideals. Moreover, we prove that this special decomposition is obtained from a cellular decomposition which is also defined in a canonical way and does not depend on the field.
A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be uniquely presented. We use the concept of gluing to construct commutative monoids with this property. Finally for some relevant families of numerical semigroups we describe the elements that are uniquely presented.
In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.
This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. Concretely, we explicitly describe how the graded minimal free resolution of those algebras is related to the combinatorics of some simplicial complexes. Our description may be interpreted as an algorithmic procedure to partially compute this resolution.
In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in \cite{fer}, we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers $ (2r,g,n) $ where $ r $ is the degree of the support, $ n \geq r $ is the dimension of the projective space containing the double curve, and $ g $ is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple $ (2r,g,n) $ is irreducible. Furthermore, we prove that the general double curve in the families associated to $ (2r,r+1,r) $ and $ (2r,1,2r-1) $ is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus $ g \leq -2 $ form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in $ \mathbb{P}^3 $ of arbitrary genus is a smooth point of the corresponding Hilbert scheme.