Narrow Results

We show that the log canonical bundle, κ, of is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of ℙ¹ × ⋯ × ℙ^n-3, permuted by the symmetric group. The homogeneous ideal of is the sum of the homogeneous ideals of these Segre embeddings.

We show that the method of moving quadrics for implicitizing surfaces in ℙ³ applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.

Given a $4$-dimensional vector subspace $U=\{f_0,\dots ,f_3\}$ of $H^0({\mathbb{P}}^1\times {\mathbb{P}}^1,{𝒪}(a,b))$, a tensor product surface, denoted by $X_U$, is the closure of the image of the rational map ${\lambda }_U:{\mathbb{P}}^1\times {\mathbb{P}}^1--\to {\mathbb{P}}^3$ determined by $U$. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of $X_U$ in ${\mathbb{P}}^3$. In this paper, we show that if $U\subseteq H^0({\mathbb{P}}^1\times {\mathbb{P}}^1,{𝒪}(a,1))$ has a finite set of $r$ basepoints in generic position, then the implicit equation of $X_U$ is determined by two syzygies of $I_U=〈f_0,\dots ,f_3〉$ in bidegrees $(a-\lceil \frac{r}{2}\rceil ,0)$ and $(a-\lfloor \frac{r}{2}\rfloor ,0)$. This result is proved by understanding the geometry of the basepoints of $U$ in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The proof techniques for the main theorem also apply when $U$ is basepoint free.

In this paper, we address the following question: for a nonzero finitely generated ideal $I$ of a multivariate polynomial ring $R[X_1,\dots ,X_n]$ over a coherent ring $R$, fixing a monomial order $<$ on $R[X_1,\dots ,X_n]$, is the trailing terms ideal $\mathrm{\text{tt}}(I)$ of $I$ (that is, the ideal generated by the trailing terms of the nonzero polynomials in $I$) finitely generated? We show that while $\mathrm{\text{tt}}(I)$ can be nonfinitely generated, it is always countably generated when the monomial order is Noetherian (graded monomial orders as instances).

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by a shifted geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Apéry set elements.

Let R₀ be a Noetherian local ring and R a standard graded R₀-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R₀, the ring R/I^k is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/I^k over R.

The Betti numbers of a graded module over the polynomial ring form a table of numerical invariants that refines the Hilbert polynomial. A sequence of papers sparked by conjectures of Boij and Söderberg have led to the characterization of the possible Betti tables up to rational multiples—that is, to the rational cone generated by the Betti tables. We will summarize this work by describing the cone and the closely related cone of cohomology tables of vector bundles on projective space, and we will give new, simpler proofs of some of the main results. We also explain some of the applications of the theory, including the one that originally motivated the conjectures of Boij and Söderberg, a proof of the Multiplicity Conjecture of Herzog, Huneke and Srinivasan.