Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Topics in Combinatorial Algebraic Geometry
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Algebra and Geometry)
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six papers in algebraic geometry –all of which have close connections to combinatorics. In Paper A we consider complete smooth toric embeddings X ↪ P^N such that for a fixed positive integer k the t-th osculating space at every point has maximal dimension if and only if t ≤ k. Our main result is that this assumption is equivalent to that X ↪ P^N is associated to a Cayley polytope of order k having every edge of length at least k. This result generalizes 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, generalizing a result of Atsushi Ito. In Paper B we introduce H-constants that measure the negativity of curves on blow-ups of surfaces. We relate these constants to the bounded negativity conjecture. Moreover we provide bounds on H-constants when restricting to curves which are a union of lines in the real or complex projective plane. In Paper C we study Gauss maps of order k for k > 1, which maps a point on a variety to its k-th osculating space at that point. Our main result is that as in the case k = 1, the higher order Gauss maps are finite on smooth varieties whose k-th osculating space is full-dimensional everywhere. Furthermore we provide convex geometric descriptions of these maps in the toric setting. In Paper D we classify fat point schemes on Hirzebruch surfaces whose initial sequence are of maximal or close to maximal length. The initial degree and initial sequence of such schemes are closely related to the famous Nagata conjecture. In Paper E we introduce the package LatticePolytopes for Macaulay2. The package extends the functionality of Macaulay2 for compuations in toric geometry and convex geometry. In Paper F we compute the Seshadri constant at a general point on smooth toric surfaces satisfying certain convex geometric assumptions on the associated polygons. Our computations relate the Seshadri constant at the general point with the jet seperation and unnormalised spectral values of the surfaces at hand. 

Abstract [sv]

Den här avhandlingen utgörs av sex artiklar inom algebraisk geometri som är nära kopplade till kombinatorik. I artikel A betraktar vi kompletta inbäddningar av glatta toriska variteter X ↪ PN sådana att för något fixt heltal k är det t-te oskulerande rummet i varje punkt av maximal dimension om och endast om t ≤ k. Vårt huvudresultat är att detta antagande är ekvivalent med att den polytop som motsvarar inbäddningen är en Cayleypolytop av ordning k, vars samtliga kanter har längd åtminstonde k. Detta resultat generaliserar en tidigare känd karaktärisering av David Perkinson. Vi visar även att ovanstående antagande är ekvivalent med antagandet att Seshadri- konstanten är lika med k i varje punkt i X. Därmed generaliserar vårt resultat ett tidigare resultat av Atsushi Ito. I artikel B introducerar vi H-konstanter, vilka mäter negativiteten av kurvor på uppblåsningar av ytor. Vi relaterar dessa konstanter till den begränsade negativitetsförmodan. Vidare erhåller vi begränsningar för konstanterna när vi enbart betraktar unioner av linjer i det reella och komplexa projektiva planet. I artikel C studerar vi Gaussavbildningen av ordning k, för k > 1, som avbildar en punkt i en varitet på det k-te oskulerande rummet i samma punkt. Vårt huvudresultat är att, i likhet med fallet k = 1, är dessa högre ordningens Gaussavbildningar ändliga på glatta variteter vars k-te oskulerande rum är fulldimensionellt överallt. Vidare ger vi konvexgeometriska beskrivningar av dessa avbildningar för toriska variteter. I artikel D klassificerar vi scheman av tjocka punkter på Hirzebruchytor vars initalsekvenser är av maximal eller nära maximal längd. Intitialgraden och initialsekvensen för sådana scheman är nära relaterade till den välkända Nagata- förmodan. I artikel E introducerar vi paketet LatticePolytopes till Macaulay2. Detta paket utökar funktionaliteten i Macaulay2 för beräkningar inom torisk och konvex geometri. I artikel F beräknar vi Seshadrikonstanten i generella punkter på glatta toriska ytor som uppfyller vissa konvexgeometriska villkor på de associerade polygonerna. Våra beräkningar koppplar samman Seshadrikonstanten i en generell punkt med jetsepareringen och det icke-normaliserade spektralvärdet hos ytorna. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. , vii, 25 p.
Series
TRITA-MAT-A, 2015:11
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-176878ISBN: 978-91-7595-734-0 (print)OAI: oai:DiVA.org:kth-176878DiVA: diva2:868497
Public defence
2015-12-04, Kollegiesalen, Brinellvägen 8, KTH, Stockholm, 09:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2010-5563Swedish Research Council, 2014-4763
Note

QC 20151112

Available from: 2015-11-12 Created: 2015-11-10 Last updated: 2016-12-15Bibliographically approved
List of papers
1. 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
2. Bounded Negativity and Arrangements of Lines
Open this publication in new window or tab >>Bounded Negativity and Arrangements of Lines
Show others...
2015 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2015, no 19, 9456-9471 p.Article in journal (Refereed) Published
Abstract [en]

The Bounded Negativity Conjecture predicts that for any smooth complex surface X there exists a lower bound for the selfintersection of reduced divisors on X. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of X. In the present note, we introduce certain constants H(X) which measure in effect the variance of the lower bounds in the birational equivalence class of X. We focus on rational surfaces and relate the value of H(ℙ^2) to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10.

Place, publisher, year, edition, pages
Oxford, England: Oxford University Press, 2015
National Category
Geometry Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176639 (URN)10.1093/imrn/rnu236 (DOI)000366499400011 ()2-s2.0-84950327309 (Scopus ID)
Funder
Swedish Research Council, NT:2010-5563
Note

QC 20160114

Available from: 2015-11-09 Created: 2015-11-09 Last updated: 2017-12-01Bibliographically approved
3. A note on higher order Gauss Maps
Open this publication in new window or tab >>A note on higher order Gauss Maps
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X, with k > 1, then the Gauss map of order k has finite fibers, unless X = P^n is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a combinatorial description of the Gauss maps of order k, its image and the general fibers. 

National Category
Geometry Algebra and Logic Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176641 (URN)
Funder
Swedish Research Council, NT:2010- 5563Swedish Research Council, NT:2014-4763
Note

QS 2015

Available from: 2015-11-09 Created: 2015-11-09 Last updated: 2015-12-14Bibliographically approved
4. The effect of points fattening on Hirzebruch surfaces
Open this publication in new window or tab >>The effect of points fattening on Hirzebruch surfaces
2015 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 288, no 5-6, 577-583 p.Article in journal (Refereed) Published
Abstract [en]

The purpose of this note is to study initial sequences of 0-dimensional subschemes of Hirzebruch surfaces and classify subschemes whose initial sequence has the minimal possible growth.

Keyword
Elementary transformations, fat points, interpolation, rational surfaces, 14C20, 14J26, 14N05, 14H20
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-166335 (URN)10.1002/mana.201400014 (DOI)000353034400008 ()2-s2.0-84927592162 (Scopus ID)
Funder
Swedish Research Council, NT:2010-5563Göran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology
Note

QC 20150508

Available from: 2015-05-08 Created: 2015-05-07 Last updated: 2017-12-04Bibliographically approved
5. LatticePolytopes: A package for computations with lattice polytopes in Macaulay2
Open this publication in new window or tab >>LatticePolytopes: A package for computations with lattice polytopes in Macaulay2
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We introduce the package LatticePolytopes for Macaulay2. The package provides methods for computations related to Cayley structures, local positivity and smoothness for lattice polytopes.

Keyword
Macaulay2, Lattice Polytopes, Toric Varieties, Local Positivity, Cayley Polytopes, Smoothness
National Category
Geometry Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176857 (URN)
Funder
Swedish Research Council, 2014-4763Swedish Research Council, 2011-5599
Note

QS 2015

Available from: 2015-11-10 Created: 2015-11-10 Last updated: 2015-11-12Bibliographically approved
6. Computing Seshardi constants on smooth toric surfaces
Open this publication in new window or tab >>Computing Seshardi constants on smooth toric surfaces
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we compute the Seshadri constants at the general point on many smooth polarized toric surfaces. We consider the case when the degree of jet separation is small or the core of the associated polygon is a line segment. Our main result is that in this case the Seshadri constant at the general point can often be determined in terms of easily computable invariants of the surfaces at hand. Lastly we consider the case that the core of the associated polygon is a point for a smooth polarized toric surface (X, L ). We show that in this case X can be constructed via consecutive equivariant blow-ups of either P^2 or P^1 x P^1. 

Keyword
Seshadri constants, toric varieties, local positivity, jet seperation, adjunction theory, fano varieties
National Category
Geometry Algebra and Logic Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-176642 (URN)
Funder
Swedish Research Council, 2014-4763
Note

QS 2015

Available from: 2015-11-09 Created: 2015-11-09 Last updated: 2015-11-12Bibliographically approved

Open Access in DiVA

Thesis_Introduction(1407 kB)137 downloads
File information
File name FULLTEXT01.pdfFile size 1407 kBChecksum SHA-512
529dca52aff9098ca37cbe0a668ffef3ea09c3b3436a45fdf66643fa659ca8e83e92aa7cd68d1f2782c64752c25b733f043da1b06ab8332b2005b822b24293ba
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Lundman, Anders
By organisation
Mathematics (Div.)
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 137 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 357 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf