This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a su... more This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I ⊂ R. It is known that there are embeddings Y ∼ = Proj k[(I e) c ] for c ≥ d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I e) c ] (which gives an arithmetic Macaulayfication of X) if and only if H 0 (Y, O Y) = k, H i (Y, O Y) = 0 for i = 1, ..., dim Y − 1, and Y is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants ε and e 0 such that k[(I e) c ] is Cohen-Macaulay for all c > d(I)e + ε and e > e 0 , and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R[(I e) c t]. If R has negative a *-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if π * O Y = O X , R i π * O Y = 0 for i > 0, and Y is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of R[(I e) c t] for all c > d(I)e + ε and e > e 0 .
We study the question of whether there is a minimum Hilbert function for double point schemes who... more We study the question of whether there is a minimum Hilbert function for double point schemes whose support is s points with generic Hilbert function. Previous work shows that this question has an affirmative answer for s ≤ 9 and for s = d 2 (for any d ∈ N). In this paper, we provide evidence in the case s = d 2 + 1, and give an affirmative answer to the question when s = 11.
Let S be a standard N r-graded algebra over a local ring A, and let M be a finitely generated Z r... more Let S be a standard N r-graded algebra over a local ring A, and let M be a finitely generated Z r-graded module over S. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules. Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.
Journal of Mathematical Analysis and Applications, 2015
Let f : R n → R be a function of class C d (d ≥ 1) such that ∂ d f ∂x d 1 ≥ λ > 0 on R n. Then th... more Let f : R n → R be a function of class C d (d ≥ 1) such that ∂ d f ∂x d 1 ≥ λ > 0 on R n. Then the following global Lojasiewicz inequality holds true: λ C d dist x, {f = 0} ∪ { ∂f ∂x 1 = 0} d ≤ |f (x)| for all x ∈ R n , where C d := d!2 2d−1 and dist(x, A) denotes the Euclidean distance from x to A. As applications of this inequality, we have the following statements: • If the sets {f = 0} and { ∂f ∂x1 = 0} are "non-asymptotic at infinity" then there exist positive constants ε and R such that λ C d dist(x, {f = 0}) d ≤ |f (x)| whenever dist(x, {f = 0}) ≤ ε and x ≥ R. • If f is a polynomial of degree d with an isolated critical point at the origin, the following effective Lojasiewicz inequality holds true c dist(x, {f = 0}) d((2d−3) n +1) 2 ≤ |f (x)| for all x ≤ r for some c > 0 and r > 0. Finally, we establish a relation between the above global Lojasiewicz inequality and the phenomenon of singularities at infinity. As a consequence, if f (x) is close to 0 then x is close to the zero set of f.
There is a one-to-one correspondence between square-free monomial ideals and clutters, which are ... more There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. In [14] it was conjectured that unmixed admissible clutters were Cohen-Macaulay. We prove that the conjecture is true for uniform clutters of heights 2 and 3, i.e., if the smallest cardinality of a minimal vertex cover of the clutter is 2 or 3. For clutters of greater height, we give a family of counterexamples to show that the conjecture fails. For unmixed admissible uniform clutters of height 4, we characterize when the Alexander dual of their edge ideals has linear quotients, and in particular, give an additional condition under which unmixed admissible uniform clutters are Cohen-Macaulay.
Proceedings of the Edinburgh Mathematical Society, 2016
We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I... more We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x0, … , xn] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.
Let (R, m) be a Noetherian local ring of dimension d > 0. Let I • = {I n } n∈N be a graded family... more Let (R, m) be a Noetherian local ring of dimension d > 0. Let I • = {I n } n∈N be a graded family of m-primary ideals in R. We examine how far off from a polynomial can the length function ℓ R (R/I n) be asymptotically. More specifically, we show that there exists a constant γ > 0 such that for all n ≥ 0, ℓ R (R/I n+1) − ℓ R (R/I n) < γn d−1 .
Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investi... more Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum I + J in A ⊗ k B in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.
Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle... more Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I s for all s ≥ 1. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I s) as s 0, and the initial value of s starting from which reg(I s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.
In a recent work [16], Kaiser, Stehlík andŠkrekovski provide a family of critically 3-chromatic g... more In a recent work [16], Kaiser, Stehlík andŠkrekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Hà and Van Tuyl [7]. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.
Connections Between Algebra, Combinatorics, and Geometry, 2014
We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial... more We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs. Dedicated to Tony Geramita, a great teacher, colleague and friend.
In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a ... more In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.
Let G be an abelian group and S be a G-graded a Noetherian algebra over a commutative ring A ⊆ S ... more Let G be an abelian group and S be a G-graded a Noetherian algebra over a commutative ring A ⊆ S 0. Let I 1 ,. .. , I s be G-homogeneous ideals in S, and let M be a finitely generated G-graded S-module. We show that the shape of non-zero G-graded Betti numbers of MI t 1 1 • • • I t s s exhibit an eventual linear behavior as the t i s get large.
The relation type question, raised by C. Huneke, asks whether for a complete equidimensional loca... more The relation type question, raised by C. Huneke, asks whether for a complete equidimensional local ring R there exists a uniform number N such that the relation type of every ideal I ⊂ R generated by a system of parameters is at most N. Wang gave a positive answer to this question when the non-Cohen-Macaulay locus of R (denoted by NCM(R)) has dimension zero. In this paper, we first present an example, due to the first author, which gives a negative answer to the question when dim NCM(R) ≥ 2. The major part of our work is to investigate the remaining situation, i.e., when dim NCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R/Ꮽ(R) is a domain, where Ꮽ(R) is the ideal generated by all homology multipliers in R. In a more general context, we also discuss many interesting properties of homology multipliers.
We survey some recent results on the minimal graded free resolution of a square-free monomial ide... more We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current state-of-the-art with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part we introduce some open questions which will hopefully inspire future research on this topic.
We survey research relating algebraic properties of powers of squarefree monomial ideals to combi... more We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.
Proceedings of the American Mathematical Society, 2009
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be co... more 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.
Let X = Proj R be a projective scheme over a field k, and let I ⊆ R be an ideal generated by form... more Let X = Proj R be a projective scheme over a field k, and let I ⊆ R be an ideal generated by forms of the same degree d. Let π : e X → X be the blowing up of X along the subscheme defined by I, and let φ : e X →X be the projection given by the divisor dE 0 − E, where E is the exceptional divisor of π and E 0 is the pullback of a general hyperplane in X. We investigate how the asymptotic linearity of the regularity and the a *-invariant of I q (for q 0) is related to invariants of fibers of φ.
An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I t) = Ass(R/I) for all t ≥ 1... more An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I t) = Ass(R/I) for all t ≥ 1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a squarefree monomial ideal I is minimally not normally torsion-free then the least power t such that I t has embedded primes is bigger than β 1 , where β 1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I t do occur when t = β 1 + 1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.
be the defining ideal of a scheme of fat points in P n 1 × • • • × P n k with support in generic ... more be the defining ideal of a scheme of fat points in P n 1 × • • • × P n k with support in generic position. When all the mi's are 1, we explicitly calculate the Castelnuovo-Mumford regularity of I. In general, if at least one mi ≥ 2, we give an upper bound for the regularity of I, which extends a result of Catalisano, Trung and Valla.
This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a su... more This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I ⊂ R. It is known that there are embeddings Y ∼ = Proj k[(I e) c ] for c ≥ d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I e) c ] (which gives an arithmetic Macaulayfication of X) if and only if H 0 (Y, O Y) = k, H i (Y, O Y) = 0 for i = 1, ..., dim Y − 1, and Y is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants ε and e 0 such that k[(I e) c ] is Cohen-Macaulay for all c > d(I)e + ε and e > e 0 , and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R[(I e) c t]. If R has negative a *-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if π * O Y = O X , R i π * O Y = 0 for i > 0, and Y is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of R[(I e) c t] for all c > d(I)e + ε and e > e 0 .
We study the question of whether there is a minimum Hilbert function for double point schemes who... more We study the question of whether there is a minimum Hilbert function for double point schemes whose support is s points with generic Hilbert function. Previous work shows that this question has an affirmative answer for s ≤ 9 and for s = d 2 (for any d ∈ N). In this paper, we provide evidence in the case s = d 2 + 1, and give an affirmative answer to the question when s = 11.
Let S be a standard N r-graded algebra over a local ring A, and let M be a finitely generated Z r... more Let S be a standard N r-graded algebra over a local ring A, and let M be a finitely generated Z r-graded module over S. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules. Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.
Journal of Mathematical Analysis and Applications, 2015
Let f : R n → R be a function of class C d (d ≥ 1) such that ∂ d f ∂x d 1 ≥ λ > 0 on R n. Then th... more Let f : R n → R be a function of class C d (d ≥ 1) such that ∂ d f ∂x d 1 ≥ λ > 0 on R n. Then the following global Lojasiewicz inequality holds true: λ C d dist x, {f = 0} ∪ { ∂f ∂x 1 = 0} d ≤ |f (x)| for all x ∈ R n , where C d := d!2 2d−1 and dist(x, A) denotes the Euclidean distance from x to A. As applications of this inequality, we have the following statements: • If the sets {f = 0} and { ∂f ∂x1 = 0} are "non-asymptotic at infinity" then there exist positive constants ε and R such that λ C d dist(x, {f = 0}) d ≤ |f (x)| whenever dist(x, {f = 0}) ≤ ε and x ≥ R. • If f is a polynomial of degree d with an isolated critical point at the origin, the following effective Lojasiewicz inequality holds true c dist(x, {f = 0}) d((2d−3) n +1) 2 ≤ |f (x)| for all x ≤ r for some c > 0 and r > 0. Finally, we establish a relation between the above global Lojasiewicz inequality and the phenomenon of singularities at infinity. As a consequence, if f (x) is close to 0 then x is close to the zero set of f.
There is a one-to-one correspondence between square-free monomial ideals and clutters, which are ... more There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. In [14] it was conjectured that unmixed admissible clutters were Cohen-Macaulay. We prove that the conjecture is true for uniform clutters of heights 2 and 3, i.e., if the smallest cardinality of a minimal vertex cover of the clutter is 2 or 3. For clutters of greater height, we give a family of counterexamples to show that the conjecture fails. For unmixed admissible uniform clutters of height 4, we characterize when the Alexander dual of their edge ideals has linear quotients, and in particular, give an additional condition under which unmixed admissible uniform clutters are Cohen-Macaulay.
Proceedings of the Edinburgh Mathematical Society, 2016
We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I... more We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x0, … , xn] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.
Let (R, m) be a Noetherian local ring of dimension d > 0. Let I • = {I n } n∈N be a graded family... more Let (R, m) be a Noetherian local ring of dimension d > 0. Let I • = {I n } n∈N be a graded family of m-primary ideals in R. We examine how far off from a polynomial can the length function ℓ R (R/I n) be asymptotically. More specifically, we show that there exists a constant γ > 0 such that for all n ≥ 0, ℓ R (R/I n+1) − ℓ R (R/I n) < γn d−1 .
Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investi... more Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum I + J in A ⊗ k B in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.
Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle... more Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I s for all s ≥ 1. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I s) as s 0, and the initial value of s starting from which reg(I s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.
In a recent work [16], Kaiser, Stehlík andŠkrekovski provide a family of critically 3-chromatic g... more In a recent work [16], Kaiser, Stehlík andŠkrekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Hà and Van Tuyl [7]. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.
Connections Between Algebra, Combinatorics, and Geometry, 2014
We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial... more We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs. Dedicated to Tony Geramita, a great teacher, colleague and friend.
In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a ... more In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.
Let G be an abelian group and S be a G-graded a Noetherian algebra over a commutative ring A ⊆ S ... more Let G be an abelian group and S be a G-graded a Noetherian algebra over a commutative ring A ⊆ S 0. Let I 1 ,. .. , I s be G-homogeneous ideals in S, and let M be a finitely generated G-graded S-module. We show that the shape of non-zero G-graded Betti numbers of MI t 1 1 • • • I t s s exhibit an eventual linear behavior as the t i s get large.
The relation type question, raised by C. Huneke, asks whether for a complete equidimensional loca... more The relation type question, raised by C. Huneke, asks whether for a complete equidimensional local ring R there exists a uniform number N such that the relation type of every ideal I ⊂ R generated by a system of parameters is at most N. Wang gave a positive answer to this question when the non-Cohen-Macaulay locus of R (denoted by NCM(R)) has dimension zero. In this paper, we first present an example, due to the first author, which gives a negative answer to the question when dim NCM(R) ≥ 2. The major part of our work is to investigate the remaining situation, i.e., when dim NCM(R) = 1. We introduce the notion of homology multipliers and show that the question has a positive answer when R/Ꮽ(R) is a domain, where Ꮽ(R) is the ideal generated by all homology multipliers in R. In a more general context, we also discuss many interesting properties of homology multipliers.
We survey some recent results on the minimal graded free resolution of a square-free monomial ide... more We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current state-of-the-art with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part we introduce some open questions which will hopefully inspire future research on this topic.
We survey research relating algebraic properties of powers of squarefree monomial ideals to combi... more We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.
Proceedings of the American Mathematical Society, 2009
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be co... more 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.
Let X = Proj R be a projective scheme over a field k, and let I ⊆ R be an ideal generated by form... more Let X = Proj R be a projective scheme over a field k, and let I ⊆ R be an ideal generated by forms of the same degree d. Let π : e X → X be the blowing up of X along the subscheme defined by I, and let φ : e X →X be the projection given by the divisor dE 0 − E, where E is the exceptional divisor of π and E 0 is the pullback of a general hyperplane in X. We investigate how the asymptotic linearity of the regularity and the a *-invariant of I q (for q 0) is related to invariants of fibers of φ.
An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I t) = Ass(R/I) for all t ≥ 1... more An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I t) = Ass(R/I) for all t ≥ 1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a squarefree monomial ideal I is minimally not normally torsion-free then the least power t such that I t has embedded primes is bigger than β 1 , where β 1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I t do occur when t = β 1 + 1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.
be the defining ideal of a scheme of fat points in P n 1 × • • • × P n k with support in generic ... more be the defining ideal of a scheme of fat points in P n 1 × • • • × P n k with support in generic position. When all the mi's are 1, we explicitly calculate the Castelnuovo-Mumford regularity of I. In general, if at least one mi ≥ 2, we give an upper bound for the regularity of I, which extends a result of Catalisano, Trung and Valla.
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