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Journal articles on the topic 'Multiprojective space'

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Author: Grafiati
Published: 7 July 2024

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1

FAVACCHIO, GIUSEPPE, and JUAN MIGLIORE. "Multiprojective spaces and the arithmetically Cohen–Macaulay property." Mathematical Proceedings of the Cambridge Philosophical Society 166, no. 3 (April 3, 2018): 583–97. http://dx.doi.org/10.1017/s0305004118000142.

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AbstractIn this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.
2

Dogan, M. Levent, Alperen A. Ergür, and Elias Tsigaridas. "On the Complexity of Chow and Hurwitz Forms." ACM Communications in Computer Algebra 57, no. 4 (December 2023): 167–99. http://dx.doi.org/10.1145/3653002.3653003.

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We consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.
3

Guardo, Elena, and Adam Van Tuyl. "Separators of points in a multiprojective space." manuscripta mathematica 126, no. 1 (February 6, 2008): 99–113. http://dx.doi.org/10.1007/s00229-008-0165-z.

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4

Guardo, Elena, and Adam Van Tuyl. "ACM sets of points in multiprojective space." Collectanea mathematica 59, no. 2 (June 2008): 191–213. http://dx.doi.org/10.1007/bf03191367.

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5

Costa, L., and R. M. Miró-Roig. "m-Blocks Collections and Castelnuovo-mumford Regularity in multiprojective spaces." Nagoya Mathematical Journal 186 (2007): 119–55. http://dx.doi.org/10.1017/s0027763000009387.

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AbstractThe main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf F on X with respect to the n-block collection B. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on ℙn and for the n-block collection Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space ℙn1x…x ℙnr with respect to a suitable n1 +…+ nr-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].
6

Miyazaki, Chikashi. "A cohomological criterion for splitting of vector bundles on multiprojective space." Proceedings of the American Mathematical Society 143, no. 4 (November 24, 2014): 1435–40. http://dx.doi.org/10.1090/s0002-9939-2014-12347-1.

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7

Miyazaki, Chikashi. "Buchsbaum criterion of Segre products of vector bundles on multiprojective space." Journal of Algebra 467 (December 2016): 47–57. http://dx.doi.org/10.1016/j.jalgebra.2016.06.037.

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8

Ballico, Edoardo. "Terracini Loci: Dimension and Description of Its Components." Mathematics 11, no. 22 (November 20, 2023): 4702. http://dx.doi.org/10.3390/math11224702.

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We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions.
9

Ballico, Edoardo. "Terracini Loci of Segre Varieties." Symmetry 14, no. 11 (November 17, 2022): 2440. http://dx.doi.org/10.3390/sym14112440.

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Fix a format (n1+1)×⋯×(nk+1), k>1, for real or complex tensors and the associated multiprojective space Y. Let V be the vector space of all tensors of the prescribed format. Let S(Y,x) denote the set of all subsets of Y with cardinality x. Elements of S(Y,x) are associated to rank 1 decompositions of tensors T∈V. We study the dimension δ(2S,Y) of the kernel at S of the differential of the associated algebraic map S(Y,x)→PV. The set T1(Y,x) of all S∈S(Y,x) such that δ(2S,Y)>0 is the largest and less interesting x-Terracini locus for tensors T∈V. Moreover, we consider the one (minimally Terracini) such that δ(2A,Y)=0 for all A⊈S. We define and study two different types of subsets of T1(Y,x) (primitive Terracini and solution sets). A previous work (Ballico, Bernardi, and Santarsiero) provided a complete classification for the cases x=2,3. We consider the case x=4 and several extremal cases for arbitrary x.
10

Francisco, Christopher A., and Adam Van Tuyl. "Some Families of Componentwise Linear Monomial Ideals." Nagoya Mathematical Journal 187 (September 2007): 115–56. http://dx.doi.org/10.1017/s0027763000025873.

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AbstractLet R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.
11

BALLICO, EDOARDO, and FRANCESCO MALASPINA. "WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES." Bulletin of the Australian Mathematical Society 84, no. 2 (July 21, 2011): 255–60. http://dx.doi.org/10.1017/s0004972711002243.

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AbstractHere we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.
12

Costa, L., and R. M. Miró-Roig. "Cohomological characterization of vector bundles on multiprojective spaces." Journal of Algebra 294, no. 1 (December 2005): 73–96. http://dx.doi.org/10.1016/j.jalgebra.2005.08.035.

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13

Favacchio, Giuseppe, Elena Guardo, and Beatrice Picone. "Special arrangements of lines: Codimension 2 ACM varieties in ℙ1 × ℙ1 × ℙ1." Journal of Algebra and Its Applications 18, no. 04 (March 25, 2019): 1950073. http://dx.doi.org/10.1142/s0219498819500737.

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In this paper, we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimension 2 arithmetically Cohen–Macaulay (ACM) varieties in [Formula: see text], called varieties of lines. We also describe their ACM property from a combinatorial algebra point of view.
14

D’Andrea, Carlos, Teresa Krick, and Martín Sombra. "Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze." Annales scientifiques de l'École normale supérieure 46, no. 4 (2013): 549–627. http://dx.doi.org/10.24033/asens.2196.

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15

Ballico, Edoardo. "Ranks on the Boundaries of Secant Varieties." New Zealand Journal of Mathematics 48 (December 31, 2018): 31–39. http://dx.doi.org/10.53733/37.

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In many cases (e.g. for many Segre or Segre-Veronese embeddings of multiprojective spaces) we prove (in characteristic 0) that a hypersurface of the b-secant variety of has X-rank > b. We prove it proving that the X-rank of a general point of the join of b − 2 copies of $X$ and the tangential variety of X is > b.
16

Coman, Dan, and James Heffers. "Lelong numbers of bidegree (1, 1) currents on multiprojective spaces." Mathematische Zeitschrift 295, no. 3-4 (November 7, 2019): 1569–82. http://dx.doi.org/10.1007/s00209-019-02427-1.

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17

Junca, Mauricio, and Mauricio Velasco. "The maximum cut problem on blow-ups of multiprojective spaces." Journal of Algebraic Combinatorics 38, no. 4 (February 15, 2013): 797–827. http://dx.doi.org/10.1007/s10801-013-0426-0.

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18

Coutinho, S. C. "Foliations of multiprojective spaces and a conjecture of Bernstein and Lunts." Transactions of the American Mathematical Society 363, no. 04 (April 1, 2011): 2125. http://dx.doi.org/10.1090/s0002-9947-2010-05230-4.

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19

Ballico, Edoardo, and Francesco Malaspina. "Regularity and cohomological splitting conditions for vector bundles on multiprojective spaces." Journal of Algebra 345, no. 1 (November 2011): 137–49. http://dx.doi.org/10.1016/j.jalgebra.2011.08.015.

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20

Censor, Yair, and Tommy Elfving. "A multiprojection algorithm using Bregman projections in a product space." Numerical Algorithms 8, no. 2 (September 1994): 221–39. http://dx.doi.org/10.1007/bf02142692.

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21

Favacchio, Giuseppe, Elena Guardo, and Juan Migliore. "On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces." Proceedings of the American Mathematical Society 146, no. 7 (February 21, 2018): 2811–25. http://dx.doi.org/10.1090/proc/13981.

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22

Costa, L., and R. M. Miró-Roig. "Corrigendum to “Cohomological characterization of vector bundles on multiprojective spaces” [J. Algebra 294 (2005) 73–96]." Journal of Algebra 319, no. 3 (February 2008): 1336–38. http://dx.doi.org/10.1016/j.jalgebra.2007.11.029.

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23

Ballico, Edoardo. "ACM curves in multiprojective spaces." Bollettino dell'Unione Matematica Italiana, February 4, 2022. http://dx.doi.org/10.1007/s40574-022-00319-7.

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24

Ballico, Edoardo. "Correction to: ACM curves in multiprojective spaces." Bollettino dell'Unione Matematica Italiana, March 28, 2022. http://dx.doi.org/10.1007/s40574-022-00321-z.

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25

Ballico, Edoardo. "Rational curves and maximal rank in multiprojective spaces." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, September 15, 2022. http://dx.doi.org/10.1007/s13366-022-00661-z.

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26

Maingi, Damian M. "Monads on multiprojective spaces and associated vector bundles." manuscripta mathematica, December 18, 2022. http://dx.doi.org/10.1007/s00229-022-01449-0.

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27

Bolognesi, Michele, Alex Massarenti, and Elena Poma. "Cox rings of blow-ups of multiprojective spaces." Collectanea Mathematica, December 7, 2023. http://dx.doi.org/10.1007/s13348-023-00428-2.

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AbstractLet $$X^{1,n}_r$$ X r 1 , n be the blow-up of $$\mathbb {P}^1\times \mathbb {P}^n$$ P 1 × P n in r general points. We describe the Mori cone of $$X^{1,n}_r$$ X r 1 , n for $$r\le n+2$$ r ≤ n + 2 and for $$r = n+3$$ r = n + 3 when $$n\le 4$$ n ≤ 4 . Furthermore, we prove that $$X^{1,n}_{n+1}$$ X n + 1 1 , n is log Fano and give an explicit presentation for its Cox ring.
28

Ballico, Edoardo. "On the Multigraded Hilbert Function of Lines and Rational Curves in Multiprojective Spaces." Vietnam Journal of Mathematics, November 9, 2021. http://dx.doi.org/10.1007/s10013-021-00537-0.

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