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Classifying Lattice Polytopes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Algebra and Geometry)
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers in toric geometry. In Paper A we provide a complete classification up to isomorphism of all smooth convex lattice 3- polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining four are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all complete embeddings of smooth toric threefolds in PN where N ≤ 15. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in PN and the remaining four are blow-ups of such toric threefolds. In Paper B we show that a complete smooth toric embedding X ↪ PN having maximal k-th osculating dimension, but not maximal (k + 1)-th osculating dimension, at every point is associated to a Cayley polytope of order k. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalising a result of Atsushi Ito. 

Abstract [sv]

Denna Licentiatuppsats utgörs av två vetenskapliga artiklar inom torisk geometri. I Paper A ger vi en komplett klassificering, upp till isomorfi, av alla 3-dimensionella glatta konvexa gitter polytoper som innehåller högst 16 gitter punkter. Totalt utgörs klassificeringen av 103 olika polytoper. Av dessa 103 polytoper är 99 stycken strikta Cayley polytoper och resterande fyra är inversa stjärnuppdelningar av Cayley polytoper. Från detta resultat härleder vi en klassificering av alla fullständiga inbäddningar av glatta toriska trefalder i PN för N ≤ 15. Återigen får vi 103 sådana inbäddningar. Av dessa är 99 projektiva fiberknippen inbäddade i PN och resterande fyra är uppblåsningar av dito. I Paper B visar vi att en fullstädig glatt torisk imbäddning X ↪ PN som i varje punkt är sådan att, det k:te oskulerande rummet har maximal dimension, men det (k + 1):a oskulerande rummet ej är av maximal dimension, är associerad till en Cayley polytop av grad k. Detta resultat generaliserar en tidigare känd klassificering av David Perkinson. Vidare visar vi att ovanstående antaganden är ekvivalenta med att anta att Seshadrikonstanten är exakt k för varje punkt på X, vilket generaliserar en tidigare klassificering av Atsushi Ito. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. , vii, 18 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 2013:04
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-134707ISBN: 978-91-7501-943-7 (print)OAI: oai:DiVA.org:kth-134707DiVA: diva2:667771
Presentation
2013-12-19, 3418, Lindstedsv. 25, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, NT:2010-5563
Note

QC 20131129

Available from: 2013-11-29 Created: 2013-11-27 Last updated: 2013-11-29Bibliographically approved
List of papers
1. A classification of smooth convex 3-polytopes with at most 16 lattice points
Open this publication in new window or tab >>A classification of smooth convex 3-polytopes with at most 16 lattice points
2013 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 37, no 1, 139-165 p.Article in journal (Refereed) Published
Abstract [en]

We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining four are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in a"(TM) (N) where Na parts per thousand currency sign15. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in a"(TM) (N) and the remaining four are blow-ups of such toric threefolds.

Keyword
Smooth, Lattice polytopes, Toric varieties, Cayley polytopes, Toric fibrations
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-116719 (URN)10.1007/s10801-012-0363-3 (DOI)000312775800008 ()2-s2.0-84871762737 (Scopus ID)
Funder
Swedish Research Council, NT:2010-5563
Note

QC 20130125

Available from: 2013-01-25 Created: 2013-01-25 Last updated: 2017-12-06Bibliographically approved
2. Local positivity of line bundles on smooth toric varieties and Cayley polytopes
Open this publication in new window or tab >>Local positivity of line bundles on smooth toric varieties and Cayley polytopes
2016 (English)In: Journal of symbolic computation, ISSN 0747-7171, E-ISSN 1095-855X, Vol. 74, 109-124 p.Article in journal (Refereed) Published
Abstract [en]

For any non-negative integer k the k-th osculating dimension at a given point x of a variety X embedded in projective space gives a measure of the local positivity of order k at that point. In this paper we show that a smooth toric embedding having maximal k-th osculating dimension, but not maximal (k + 1)-th osculating dimension, at every point is associated to a Cayley polytope of order k. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalising a result of Atsushi Ito. 

Keyword
osculating space, Seshadri consant, k-jet ampleness, toric variety, Cayley polytope, lattice polytope
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-134705 (URN)10.1016/j.jsc.2015.05.007 (DOI)000366794100006 ()2-s2.0-84948718076 (Scopus ID)
Funder
Swedish Research Council, NT:2010-5563
Note

QC 20160121

Available from: 2013-11-27 Created: 2013-11-27 Last updated: 2017-12-06Bibliographically approved

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