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Boij, M. & Lundqvist, S. (2023). A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms. Algebra & Number Theory, 17(1), 111-126
Öppna denna publikation i ny flik eller fönster >>A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms
2023 (Engelska)Ingår i: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 17, nr 1, s. 111-126Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miro-Roig, and Nagel.

Ort, förlag, år, upplaga, sidor
Mathematical Sciences Publishers, 2023
Nyckelord
powers of linear forms, general linear forms, almost complete intersections, weak Lefschetz property, inverse system, Hilbert series
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-327422 (URN)10.2140/ant.2023.17.111 (DOI)000968563600005 ()2-s2.0-85148596480 (Scopus ID)
Anmärkning

QC 20230530

Tillgänglig från: 2023-05-30 Skapad: 2023-05-30 Senast uppdaterad: 2023-05-30Bibliografiskt granskad
Boij, M., Migliore, J., Miró-Roig, R. M. & Nagel, U. (2023). On the weak lefschetz property for height four equigenerated complete intersections. Transactions of the American Mathematical Society Series B, 10(35), 1254-1286
Öppna denna publikation i ny flik eller fönster >>On the weak lefschetz property for height four equigenerated complete intersections
2023 (Engelska)Ingår i: Transactions of the American Mathematical Society Series B, E-ISSN 2330-0000, Vol. 10, nr 35, s. 1254-1286Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P3, and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3, 4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P3.

Ort, förlag, år, upplaga, sidor
American Mathematical Society (AMS), 2023
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-337430 (URN)10.1090/btran/163 (DOI)2-s2.0-85171770436 (Scopus ID)
Anmärkning

QC 20231003

Tillgänglig från: 2023-10-03 Skapad: 2023-10-03 Senast uppdaterad: 2023-10-03Bibliografiskt granskad
Boij, M., Migliore, J., Miro-Roig, R. M. & Nagel, U. (2022). Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms. Algebra & Number Theory, 16(1), 155-178
Öppna denna publikation i ny flik eller fönster >>Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms
2022 (Engelska)Ingår i: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 16, nr 1, s. 155-178Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the strong Lefschetz property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the strong Lefschetz property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

Ort, förlag, år, upplaga, sidor
Mathematical Sciences Publishers, 2022
Nyckelord
Waring rank, cactus rank, symmetric forms, strong Lefschetz property, Macaulay duality, minimal free resolution, power sum decomposition, Gorenstein algebra
Nationell ämneskategori
Algebra och logik
Identifikatorer
urn:nbn:se:kth:diva-310225 (URN)10.2140/ant.2022.16.155 (DOI)000761261600004 ()2-s2.0-85126532132 (Scopus ID)
Anmärkning

QC 20220325

Tillgänglig från: 2022-03-25 Skapad: 2022-03-25 Senast uppdaterad: 2022-06-25Bibliografiskt granskad
Boij, M. & Teitler, Z. (2020). A bound for the Waring rank of the determinant via syzygies. Linear Algebra and its Applications, 587, 195-214
Öppna denna publikation i ny flik eller fönster >>A bound for the Waring rank of the determinant via syzygies
2020 (Engelska)Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 587, s. 195-214Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We show that the Waring rank of the 3 x 3 determinant, previously known to be between 14 and 18, is at least 15. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the symmetric cactus rank of the 3 x 3 permanent is at least 14.

Ort, förlag, år, upplaga, sidor
ELSEVIER SCIENCE INC, 2020
Nyckelord
Waring rank, Symmetric rank, Symmetric cactus rank, Determinants, Permanents, Syzygies
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-269455 (URN)10.1016/j.laa.2019.11.007 (DOI)000514011600009 ()2-s2.0-85074761605 (Scopus ID)
Anmärkning

QC 20200310

Tillgänglig från: 2020-03-10 Skapad: 2020-03-10 Senast uppdaterad: 2022-06-26Bibliografiskt granskad
Altafi, N. & Boij, M. (2020). The weak Lefschetz property of equigenerated monomial ideals. Journal of Algebra, 556, 136-168
Öppna denna publikation i ny flik eller fönster >>The weak Lefschetz property of equigenerated monomial ideals
2020 (Engelska)Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 556, s. 136-168Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We determine a sharp lower bound for the Hilbert function in degree d of a monomial algebra failing the weak Lefschetz property over a polynomial ring with n variables and generated in degree d, for any d≥2 and n≥3. We consider artinian ideals in the polynomial ring with n variables generated by homogeneous polynomials of degree d invariant under an action of the cyclic group Z/dZ, for any n≥3 and any d≥2. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.

Ort, förlag, år, upplaga, sidor
Academic Press, 2020
Nyckelord
Group actions, Monomial ideals, Weak Lefschetz property
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-276292 (URN)10.1016/j.jalgebra.2020.02.020 (DOI)000530229500006 ()2-s2.0-85082688423 (Scopus ID)
Anmärkning

QC 20200617

Tillgänglig från: 2020-06-17 Skapad: 2020-06-17 Senast uppdaterad: 2024-03-18Bibliografiskt granskad
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
Öppna denna publikation i ny flik eller fönster >>The minimal resolution conjecture on a general quartic surface in P-3
2019 (Engelska)Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 223, nr 4, s. 1456-1471Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Elsevier, 2019
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-240685 (URN)10.1016/j.jpaa.2018.06.014 (DOI)000452581900006 ()2-s2.0-85048877738 (Scopus ID)
Anmärkning

QC 20190110

Tillgänglig från: 2019-01-10 Skapad: 2019-01-10 Senast uppdaterad: 2022-06-26Bibliografiskt granskad
Boij, M. & Conca, A. (2018). On Fröberg-Macaulay conjectures for algebras. Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 50, 139-147
Öppna denna publikation i ny flik eller fönster >>On Fröberg-Macaulay conjectures for algebras
2018 (Engelska)Ingår i: Rendiconti dell'Istituto di Matematica dell'Università di Trieste, ISSN 0049-4704, E-ISSN 2464-8728, Vol. 50, s. 139-147Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
EUT Edizioni Universita di Trieste, 2018
Nyckelord
Hilbert functions, Macaulay theorem
Nationell ämneskategori
Matematisk analys
Identifikatorer
urn:nbn:se:kth:diva-247420 (URN)10.13137/2464-8728/22433 (DOI)2-s2.0-85060528366 (Scopus ID)
Anmärkning

QC20190502

Tillgänglig från: 2019-05-03 Skapad: 2019-05-03 Senast uppdaterad: 2022-06-26Bibliografiskt granskad
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
Öppna denna publikation i ny flik eller fönster >>Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections
2018 (Engelska)Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 495, s. 1-14Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Academic Press, 2018
Nyckelord
Fröberg's conjecture, Generic forms, Hilbert series, Weak Lefschetz property
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-218917 (URN)10.1016/j.jalgebra.2017.11.001 (DOI)000418106900001 ()2-s2.0-85033590818 (Scopus ID)
Forskningsfinansiär
Vetenskapsrådet, VR2013-4545
Anmärkning

QC 20171201

Tillgänglig från: 2017-12-01 Skapad: 2017-12-01 Senast uppdaterad: 2022-06-26Bibliografiskt granskad
Boij, M., Migliore, J., Miró-Roig, R. M. & Nagel, U. (2018). The non-Lefschetz locus. Journal of Algebra, 505, 288-320
Öppna denna publikation i ny flik eller fönster >>The non-Lefschetz locus
2018 (Engelska)Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 505, s. 288-320Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Academic Press, 2018
Nyckelord
Artinian algebra, Complete intersection, Gorenstein algebra, Weak Lefschetz property
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-227523 (URN)10.1016/j.jalgebra.2018.03.006 (DOI)000431746300012 ()2-s2.0-85044454533 (Scopus ID)
Anmärkning

QC 20180515

Tillgänglig från: 2018-05-15 Skapad: 2018-05-15 Senast uppdaterad: 2022-06-26Bibliografiskt granskad
Boij, M. & Smith, G. G. (2015). Cones of Hilbert Functions. International mathematics research notices (20), 10314-10338
Öppna denna publikation i ny flik eller fönster >>Cones of Hilbert Functions
2015 (Engelska)Ingår i: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, nr 20, s. 10314-10338Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Oxford University Press, 2015
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:kth:diva-180386 (URN)10.1093/imrn/rnu265 (DOI)000366500400015 ()2-s2.0-84948389776 (Scopus ID)
Anmärkning

QC 20160114

Tillgänglig från: 2016-01-14 Skapad: 2016-01-13 Senast uppdaterad: 2022-06-23Bibliografiskt granskad
Organisationer
Identifikatorer
ORCID-id: ORCID iD iconorcid.org/0000-0002-9961-383X

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