Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P^3 defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph to each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.Comment: 15 pages; minor revisions to v. 1 to improve clarity; to appear in JPA

Borel generators

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets.Comment: 23 pages, 2 figures; very minor changes in v2. To appear in J. Algebr

Generalizing the Borel property

We introduce the notion of Q-Borel ideals: ideals which are closed under the Borel moves arising from a poset Q. We study decompositions and homological properties of these ideals, and offer evidence that they interpolate between Borel ideals and arbitrary monomial ideals.Comment: 19 pages, 1 figur

On the componentwise linearity and the minimal free resolution of a tetrahedral curve

A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this work by showing that they are "almost always" componentwise linear, i.e. their homogeneous ideals have the property that for any d, the degree d component of the ideal generates a new ideal whose minimal free resolution is linear. The one type of exception is clearly spelled out and studied as well. The main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves, and the connection to the tetrahedral curves that are minimal in their even liaison classes. With this preparation, we also describe the minimal free resolution of a tetrahedral curve, and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution. Finally, we begin the study of the generic initial ideal (gin) of a tetrahedral curve. We produce the gin for arithmetically Cohen-Macaulay tetrahedral curves and for minimal arithmetically Buchsbaum tetrahedral curves, and we show how to obtain it for any non-minimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class.Comment: 31 pages; v2 has very minor changes: fixed typos, added Remark 4.2 and char. zero hypothesis to 5.2, and reworded 5.5. To appear, J. Algebr

Splittings of monomial ideals

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.Comment: minor changes: added Cor. 3.10 and some references. To appear in Proc. Amer. Math. So