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What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws., Comment: 12 pages

We show that a Buchsbaum simplicial complex of small codimension must have large depth. More generally, we achieve a similar result for ${\rm CM}_t$ simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more precise results in the codimension 2 case. Along the paper, we show that the ${\rm CM}_t$ property is a topological invariant of a simplicial complex., 9 pages, 1 figure

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our work are: (1) Every almost-closed strongly-connected d-dimensional simplicial complex is traceable. (This extends the well-known result "unit-interval connected graphs are traceable".) (2) Every almost-closed d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian. (This extends the fact that "unit-interval 2-connected graphs are Hamiltonian".) (3) Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Groebner basis for a diagonal term order which is compatible with the traceability of the complex. (This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa-Vandebogert.) (4) Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution. (This extends the result by Kiani and Saeedi-Madani that "only the complete graph has a binomial edge ideal with linear resolution".) (5) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure., 41 pages, 5 figures; improved and extended version, with Main Theorem V, Lemma 48, and Corollary 81 added, plus minor corrections and strengthenings

In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform some numerical experiments. As a major byproduct, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.

International audience; We answer several natural questions which arise from the recent paper [MP] of McCullough and Peeva providing counterexamples to the Eisenbud-Goto Regularity Conjecture. We give counterexamples using Rees algebras, and also construct counterexamples that do not rely on the Mayr-Meyer construction. Furthermore, examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given. Using a result of Ananyan-Hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone.

During the conference held in 2017 in Minneapolis for his 60th birthday, Gennady Lyubeznik proposed the following problem: Find a complete local domain and an element in it having three minimal primes such that the sum of any two of them has height 2 and the sum of the three of them has height 4. In this note this beautiful problem will be discussed, and will be shown that the principle leading to the fact that such a ring cannot exist is false. The specific problem, though, remains open, A relevant typo fixed!

We begin the study of the notion of diameter of an ideal I ⊂ S = k [ x 1 , … , x n ] , an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals satisfying diam ( I ) ≤ height ( I ) , such as: quadratic radical ideals such that S / I is Gorenstein and height ( I ) ≤ 4 , or ideals admitting a square-free complete intersection initial ideal.

We show that for every positive integer R there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to R. Such examples can not be found among Gorenstein ideals since the regularity of their quotients is at most four. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity grows at most doubly logarithmically in the number of variables.

We build a new bridge between geometric group theory and commutative algebra by showing that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection and techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay., Oberwolfach Preprints;2017,15

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen–Macaulay module.

A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of when a standard graded strongly $F$-regular ring is Gorenstein, in terms of an $F$-pure threshold. We prove this conjecture under the additional hypothesis that the anti-canonical cover of the ring is Noetherian. Moreover, under this hypothesis on the anti-canonical cover, we give a similar criterion for when a normal $F$-pure (resp. log canonical) singularity is quasi-Gorenstein, in terms of an $F$-pure (resp. log canonical) threshold.

Given an algebraic set X P n , its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne’s connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne’s result: (1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo{Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski’s theorem for simplicial polytopes.) (2) If X is an arrangement of lines no three of which meet in the same point, and X is canonically embedded in P n , then the diameter of the graph G(X) is not larger than codimPn X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for ag normal simplicial complexes.)

We characterize the irreducible representations of the general linear group GL(V) that have multiplicity 1 in the direct sum of all Schur modules of a given exterior power of V. These have come up in connection with the relations of the lower order minors of a generic matrix. We show that the minimal relations conjectured by Bruns, Conca and Varbaro are exactly those coming from partitions of single exterior type.

It is shown that the h-vectors of Stanley-Reisner rings of three classes of matroids are pure O-sequences. The classes are (a) matroids that are truncations of other matroids, or more generally of Cohen-Macaulay complexes, (b) matroids whose dual is (rank + 2)-partite, and (c) matroids of Cohen-Macaulay type at most five. Consequences for the computational search for a counterexample to a conjecture of Stanley are discussed., Comment: 16 pages, v2: various small improvements, accepted by Bulletin of the London Math. Society

In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a well-known theorem of Peskine and Szpiro. As a corollary, we give new examples of prime ideals that are not set-theoretically Cohen-Macaulay., 6 pages. Corrected some typos and added details in Example 3.9. Added Example 2.3 and rearranged the proof of Proposition 3.1. To appear in Compositio Math

A recent result on the configuration of the irreducible components of a projective scheme will be described.

We show that in arithmetically-Gorenstein line arrangements with only planar singularities, each line intersects the same number of other lines. This number has an algebraic interpretation: it is the Castelnuovo-Mumford regularity of the coordinate ring of the arrangement. We also prove that every (d-1)-dimensional simplicial complex whose 0-th and 1-st homologies are trivial is the nerve complex of a suitable d-dimensional standard graded algebra of depth $\ge 3$. This provides the converse of a recent result by Katzman, Lyubeznik and Zhang., Comment: 13 pages; minor modifications throughout; to appear in J Pure Applied Algebra

In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger than the projective dimension of S/I and (iii) the K-algebra generated by some generators of I is a direct summand of S, then f(k) is constant. When I is a square-free monomial ideal, the above criterion includes as special cases all the results of a recent paper by Herzog and Vladoiu. In this combinatorial setting there is a chance that the converse of the above fact holds true., Comment: The title has been changed and other minor changes have been done. The paper will appear in Journal of Algebra

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs., Comment: 23 pages

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen--Macaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo--Mumford regularity. If $X \subset \mathbb{P}^n$ is an arithmetically Gorenstein projective scheme of regularity $r+1$, and if every irreducible component of $X$ has regularity $\le r'$, we show that the dual graph of $X$ is $\lfloor{\frac{r+r'-1}{r'}}\rfloor$-connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph of a suitable arithmetically Cohen-Macaulay projective curve of regularity $\le 3$, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences of our work are: (1) Any graph is the Hochster-Huneke graph of a complete equidimensional local ring. (This answers a question by Sather-Wagstaff and Spiroff.) (2) The regularity of a curve is not larger than the sum of the regularities of its primary components., Comment: Added Remark 1.1 and Example 4.3; improved exposition

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Bjorner. In this note, we provide a unifying approach which allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley--Reisner rings., Comment: 4 pages, minor changes

We will see the proof of the lemma that allowed Caviglia, Constantinescu and myself in (Isr. J. Math. 204, 469–475 (2014)) to show a version of a conjecture by Kalai. We will also discuss possible consequences of the lemma on the study of dual graphs of flag complexes.

In this paper we partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley’s conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.

In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal., Comment: Minor changes, especially the addition of reference [Ya15]

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that $h$-vectors of flag Cohen-Macaulay simplicial complexes are $h$-vectors of Cohen-Macaulay balanced simplicial complexes., Comment: 3 pages

This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL ( V ) × GL ( W ) -invariant ideal in S = Sym ( V ⊗ W ⁎ ) , where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal. Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we compute the test ideals (and so the F-pure thresholds) of any sum of products of determinantal ideals. Not all the invariant ideals are as the latter (not even in characteristic 0), but they are up to integral closure, and this is enough to reach our goals. The results concerning the test ideals are obtained as a consequence of general results holding true in a special situation. Within such framework fall determinantal objects of a generic matrix, as well as of a symmetric matrix and of a skew-symmetric one. Similar results are thus deduced for the GL ( V ) -invariant ideals in Sym ( Sym 2 V ) and in Sym ( ⋀ 2 V ) . (Monomial ideals also fall in this framework, thus we recover Howald's formula for their multiplier ideals and, more generally, Hara–Yoshida's formula for their test ideals.) In the proof, we introduce the notion of “floating test ideals”, a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities. As will be shown, products of determinantal ideals, and by passing to characteristic 0 ideals generated by a single Schur module, have this property.

The $j$-multiplicity plays an important role in the intersection theory of St\"uckrad-Vogel cycles, while recent developments confirm the connections between the $\epsilon$-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the $j$-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$- and $\epsilon$-multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory., Comment: 27 pages; to appear in Proc. London Math. Soc

We define the notion of componentwise regularity and study some of its basic properties. We prove an analogue, when working with weight orders, of Buchberger's criterion to compute Gr\"obner bases; the proof of our criterion relies on a strengthening of a lifting lemma of Buchsbaum and Eisenbud. This criterion helps us to show a stronger version of Green's crystallization theorem in a quite general setting, according to the componentwise regularity of the initial object. Finally we show a necessary condition, given a submodule $M$ of a free one over the polynomial ring and a weight such that $in(M)$ is componentwise linear, for the existence of an $i$ such that $\beta_i(M)=\beta_i(in(M))$., Comment: Minor changes to the introduction. Added Corollary 5.6. Strengthened conclusion of Theorem 5.7

An ideal I in a polynomial ring S has linear powers if all the powers I^k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers., Final version, minor changes, to appear in Journal f\"ur die reine und angewandte Mathematik (Crelles Journal)

The $F$-pure threshold is a numerical invariant of prime characteristic singularities, that constitutes an analogue of the log canonical thresholds in characteristic zero. We compute the $F$-pure thresholds of determinantal ideals, i.e., of ideals generated by the minors of a generic matrix.

It is well-known that the Pl\"ucker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations between minors of a (non-maximal) fixed size. We will exhibit minimal relations in degrees 2 (non-Pl\"ucker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group., Comment: Final version, minor changes, to appear in Advances in Mathematics

In this paper we discuss the problem of characterizing the Cohen-Macaulay property of certain families of monomial ideals with fixed radical. More precisely, we consider generically complete intersection monomial ideals whose radical corresponds to special classes of simplicial complexes., To appear in Comm. Alg

We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological dimension of certain schemes. In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme, in the characteristic-zero-case and under a smoothness assumption. Furthermore, we show that a relationship between depth and cohomological dimension discovered by Peskine and Szpiro in positive characteristic holds true also in characteristic-zero up to dimension three., 20 pages, to appear on Transaction of the AMS