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Publications (10 of 16) Show all publications
Boij, M., Migliore, J., Miro-Roig, R. M. & Nagel, U. (2019). The minimal resolution conjecture on a general quartic surface in P-3. Journal of Pure and Applied Algebra, 223(4), 1456-1471
Open this publication in new window or tab >>The minimal resolution conjecture on a general quartic surface in P-3
2019 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 223, no 4, p. 1456-1471Article in journal (Refereed) Published
Abstract [en]

Mustata has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in P-3 this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.

Place, publisher, year, edition, pages
Elsevier, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-240685 (URN)10.1016/j.jpaa.2018.06.014 (DOI)000452581900006 ()2-s2.0-85048877738 (Scopus ID)
Note

QC 20190110

Available from: 2019-01-10 Created: 2019-01-10 Last updated: 2019-01-10Bibliographically approved
Boij, M. & Conca, A. (2018). On Fröberg-Macaulay conjectures for algebras. Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 50, 139-147
Open this publication in new window or tab >>On Fröberg-Macaulay conjectures for algebras
2018 (English)In: Rendiconti dell'Istituto di Matematica dell'Università di Trieste, ISSN 0049-4704, E-ISSN 2464-8728, Vol. 50, p. 139-147Article in journal (Refereed) Published
Abstract [en]

Macaulay's theorem and Fröberg's conjecture deal with the Hilbert function of homogeneous ideals in polynomial rings over a field K. In this short note we present some questions related to variants of Macaulay's theorem and Fröberg's conjecture for K-subalgebras of polynomial rings.

Place, publisher, year, edition, pages
EUT Edizioni Universita di Trieste, 2018
Keywords
Hilbert functions, Macaulay theorem
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-247420 (URN)10.13137/2464-8728/22433 (DOI)2-s2.0-85060528366 (Scopus ID)
Note

QC20190502

Available from: 2019-05-03 Created: 2019-05-03 Last updated: 2019-05-03Bibliographically approved
Boij, M., Fröberg, R. & Lundqvist, S. (2018). Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections. Journal of Algebra, 495, 1-14
Open this publication in new window or tab >>Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 495, p. 1-14Article in journal (Refereed) Published
Abstract [en]

Given an ideal I=(f1,…,fr) in C[x1,…,xn] generated by forms of degree d, and an integer k>1, how large can the ideal Ik be, i.e., how small can the Hilbert function of C[x1,…,xn]/Ik be? If r≤n the smallest Hilbert function is achieved by any complete intersection, but for r>n, the question is in general very hard to answer. We study the problem for r=n+1, where the result is known for k=1. We also study a closely related problem, the Weak Lefschetz property, for S/Ik, where I is the ideal generated by the d'th powers of the variables.

Place, publisher, year, edition, pages
Academic Press, 2018
Keywords
Fröberg's conjecture, Generic forms, Hilbert series, Weak Lefschetz property
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-218917 (URN)10.1016/j.jalgebra.2017.11.001 (DOI)000418106900001 ()2-s2.0-85033590818 (Scopus ID)
Funder
Swedish Research Council, VR2013-4545
Note

QC 20171201

Available from: 2017-12-01 Created: 2017-12-01 Last updated: 2018-12-17Bibliographically approved
Boij, M., Migliore, J., Miró-Roig, R. M. & Nagel, U. (2018). The non-Lefschetz locus. Journal of Algebra, 505, 288-320
Open this publication in new window or tab >>The non-Lefschetz locus
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 505, p. 288-320Article in journal (Refereed) Published
Abstract [en]

We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra A we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms ℓ give maximal rank for all the multiplication maps ×ℓ:[A]i⟶[A]i+1. This is a Zariski open set and its complement is the non-Lefschetz locus. For monomial complete intersections, we completely describe the non-Lefschetz locus. For general complete intersections of codimension three and four we prove that the non-Lefschetz locus has the expected codimension, which in particular means that it is empty in a large family of examples. For general Gorenstein algebras of codimension three with a given Hilbert function, we prove that the non-Lefschetz locus has the expected codimension if the first difference of the Hilbert function is of decreasing type. For completeness we also give a full description of the non-Lefschetz locus for artinian algebras of codimension two.

Place, publisher, year, edition, pages
Academic Press, 2018
Keywords
Artinian algebra, Complete intersection, Gorenstein algebra, Weak Lefschetz property
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-227523 (URN)10.1016/j.jalgebra.2018.03.006 (DOI)2-s2.0-85044454533 (Scopus ID)
Note

QC 20180515

Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-05-15Bibliographically approved
Boij, M. & Smith, G. G. (2015). Cones of Hilbert Functions. International mathematics research notices (20), 10314-10338
Open this publication in new window or tab >>Cones of Hilbert Functions
2015 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 20, p. 10314-10338Article in journal (Refereed) Published
Abstract [en]

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree 0, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded alpha-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.

Place, publisher, year, edition, pages
Oxford University Press, 2015
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-180386 (URN)10.1093/imrn/rnu265 (DOI)000366500400015 ()2-s2.0-84948389776 (Scopus ID)
Note

QC 20160114

Available from: 2016-01-14 Created: 2016-01-13 Last updated: 2017-11-30Bibliographically approved
Boij, M., Migliore, J., Miró-Roig, R. M., Nagel, U. & Zanello, F. (2014). On the Weak Lefschetz Property for artinian Gorenstein algebras of codimension three. Journal of Algebra, 403, 48-68
Open this publication in new window or tab >>On the Weak Lefschetz Property for artinian Gorenstein algebras of codimension three
Show others...
2014 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 403, p. 48-68Article in journal (Refereed) Published
Abstract [en]

We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1, 3, 6, 6, 3, 1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from P2 to P3, and Hesse configurations in P2.

Keywords
Artinian algebra, Gorenstein algebra, Hesse configuration, Primary, Secondary, Weak Lefschetz Property
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-142983 (URN)10.1016/j.jalgebra.2014.01.003 (DOI)000332349700004 ()2-s2.0-84893155346 (Scopus ID)
Note

QC 20140314

Available from: 2014-03-14 Created: 2014-03-14 Last updated: 2017-12-05Bibliographically approved
Boij, M. & Söderberg, J. (2012). Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case. Algebra & Number Theory, 6(3), 437-454
Open this publication in new window or tab >>Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case
2012 (English)In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 6, no 3, p. 437-454Article in journal (Refereed) Published
Abstract [en]

We use results of Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination of Betti diagrams of modules with a pure resolution. This implies the multiplicity conjecture of Herzog, Huneke, and Srinivasan for modules that are not necessarily Cohen-Macaulay and also implies a generalized version of these inequalities. We also give a combinatorial proof of the convexity of the simplicial fan spanned by pure diagrams.

Keywords
graded modules, Betti numbers, multiplicity conjecture
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-100179 (URN)10.2140/ant.2012.6.437 (DOI)000306191600002 ()2-s2.0-84863800231 (Scopus ID)
Note
QC 20120806Available from: 2012-08-06 Created: 2012-08-06 Last updated: 2017-12-07Bibliographically approved
Boij, M., Migliore, J. C., Miro-Roig, R. M., Nagel, U. & Zanello, F. (2012). On the Shape of a Pure O-sequence. Memoirs of the American Mathematical Society, 218(1024), 1-+
Open this publication in new window or tab >>On the Shape of a Pure O-sequence
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2012 (English)In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 218, no 1024, p. 1-+Article in journal (Refereed) Published
Abstract [en]

A monomial order ideal is a finite collection X of (monic) monomials such that, whenever M is an element of X and N divides M, then N is an element of X. Hence X is a poset, where the partial order is given by divisibility. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, (h) under bar = (h(0) = 1, h(1), ..., h(e)), counting the monomials of X in each degree. Equivalently, pure O-sequences can be characterized as the f-vectors of pure multicomplexes, or, in the language of commutative algebra, as the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of the early works of Stanley's in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their f-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, which makes an extensive use of both algebraic and combinatorial techniques, in particular includes: (i) A characterization of the first half of a pure O-sequence, which yields the exact converse to a g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, a natural bijection between integer partitions and type 1 pure O-sequences, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) A study of the Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) full characterization; (v) A pithy connection of the ICP with Stanley's conjecture on the h-vectors of matroid complexes; (vi) A more specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 over a field of characteristic zero. As an immediate corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over an arbitrary field). (vii) An analysis, from a commutative algebra viewpoint, of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras. (viii) Some observations about pure f-vectors, an important special case of pure O-sequences.

Keywords
Pure O-sequence, Artinian algebra, monomial algebra, unimodality, differentiable O-sequence, level algebra, Gorenstein algebra, enumeration, interval conjecture, g-element, weak Lefschetz property, strong Lefschetz property, matroid sirnplicial complex, Macaulay's inverse system
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-99067 (URN)000305504300001 ()
Note
QC 20120719Available from: 2012-07-19 Created: 2012-07-13 Last updated: 2017-12-07Bibliographically approved
Boij, M., Carlini, E. & Geramita, A. V. (2011). Monomials as sums of powers: The real binary case. Proceedings of the American Mathematical Society, 139(9), 3039-3043
Open this publication in new window or tab >>Monomials as sums of powers: The real binary case
2011 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 139, no 9, p. 3039-3043Article in journal (Refereed) Published
Abstract [en]

We generalize an example, due to Sylvester, and prove that any monomial of degree d in R[x(0), x(1)], which is not a power of a variable, cannot be written as a linear combination of fewer than d powers of linear forms.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-39013 (URN)10.1090/S0002-9939-2011-11018-9 (DOI)000293719900002 ()2-s2.0-79959319405 (Scopus ID)
Available from: 2011-09-06 Created: 2011-09-06 Last updated: 2017-12-08Bibliographically approved
Boij, M. & Fløystad, G. (2011). The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two. In: Floystad, G; Johnsen, T; Knutsen, AL (Ed.), Combinatorial Aspects Of Commutative Algebra And Algebraic Geometry: The Abel Symposium 2009. Paper presented at Abel Symposium on Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Voss, Norway, June 01-04, 2009 (pp. 1-16). Springer Berlin/Heidelberg, 6
Open this publication in new window or tab >>The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two
2011 (English)In: Combinatorial Aspects Of Commutative Algebra And Algebraic Geometry: The Abel Symposium 2009 / [ed] Floystad, G; Johnsen, T; Knutsen, AL, Springer Berlin/Heidelberg, 2011, Vol. 6, p. 1-16Conference paper, Published paper (Refereed)
Abstract [en]

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences, p and q, of these total degrees are relatively prime, the extremal rays are parametrized by order ideals in N-2 contained in the region px + qy < (p - 1)(q - 1). We also consider some examples concerning artinian modules of codimension three.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2011
Series
Abel Symposia ; 6
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-100984 (URN)10.1007/978-3-642-19492-4_1 (DOI)000303079200001 ()2-s2.0-84861343940 (Scopus ID)978-3-642-19491-7 (ISBN)
Conference
Abel Symposium on Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Voss, Norway, June 01-04, 2009
Note
QC 20120822Available from: 2012-08-22 Created: 2012-08-22 Last updated: 2012-08-22Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-9961-383X

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