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Topics in computation, numerical methods and algebraic  geometry
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers.

In Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes.

In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism.

Paper IV is a study of curves on certain special quartic surfaces in projective 3-space. The surfaces are invariant under the action of a finite group called the level (2,2) Heisenberg group. In the paper, we determine the Picard group of a very general member of this family of quartics. We have found that the general Heisenberg invariant quartic contains 320 smooth conics and we prove that in the very general case, this collection of conics generates the Picard group.

Place, publisher, year, edition, pages
Stockholm: KTH , 2010. , v, 20 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 10:13
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-25941ISBN: 978-91-7415-770-3 (print)OAI: oai:DiVA.org:kth-25941DiVA: diva2:360978
Public defence
2010-11-29, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20101115Available from: 2010-11-15 Created: 2010-11-05 Last updated: 2010-11-15Bibliographically approved
List of papers
1. Chern numbers of smooth varieties via homotopy continuation and intersection theory
Open this publication in new window or tab >>Chern numbers of smooth varieties via homotopy continuation and intersection theory
2011 (English)In: Journal of symbolic computation, ISSN 0747-7171, E-ISSN 1095-855X, Vol. 46, no 1, 23-33 p.Article in journal (Refereed) Published
Abstract [en]

Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in Pn. We illustrate the approach through several worked examples.

Keyword
Homotopy continuation, Numerical algebraic geometry, Polynomial system, Linear system, Linkage, Curve, Surface
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-26106 (URN)10.1016/j.jsc.2010.06.026 (DOI)000284391300002 ()2-s2.0-77958151034 (Scopus ID)
Funder
Swedish Research Council, NT:2006-3539Knut and Alice Wallenberg Foundation
Note
QC 20101115Available from: 2010-11-15 Created: 2010-11-15 Last updated: 2017-12-12Bibliographically approved
2. Computing intersection numbers of Chern classes
Open this publication in new window or tab >>Computing intersection numbers of Chern classes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let Z Pr be a smooth variety of dimension n and let c0, . . . , cn be the Chern classes of Z. We present an algorithm to compute the degree of any monomial in {c0, . . . , cn}. The method is based on intersection theory and may be implemented as a numeric, symbolic, or as a numeric/symbolic hybrid algorithm.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-26109 (URN)
Note
QC 20101115Available from: 2010-11-15 Created: 2010-11-15 Last updated: 2010-11-15Bibliographically approved
3. Algebraic C*-actions and inverse kinematics
Open this publication in new window or tab >>Algebraic C*-actions and inverse kinematics
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Let X be a smooth quadric of dimension 2m in P2m+1C and let Y,Z X be subvarietiesboth of dimension m which intersect transversely. In this paper we give an algorithm forcomputing the intersection points of Y  Z based on a homotopy method. The homotopyis constructed using a C*-action on X whose fixed points are isolated, which inducesthe so-called Bialynicki-Birula decompositions of X into locally closed invariant subsets.Notably, the homotopy has the optimal number of solution paths. As an applicationwe present a new solution to the inverse kinematics problem of a general six-revoluteserial-link manipulator.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-26112 (URN)
Note
QC 20101115Available from: 2010-11-15 Created: 2010-11-15 Last updated: 2010-11-15Bibliographically approved
4. Curves on Heisenberg invariant quartic surfaces in projective 3-space
Open this publication in new window or tab >>Curves on Heisenberg invariant quartic surfaces in projective 3-space
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper is about the family of smooth quartic surfaces X P3 that are invariant under the Heisenberg group H2,2. For a very generic X, we show that the Picard number of X is 16 and determine its Picard group. It turns out that a very generic X contains 320 irreducible conics which generate the Picard group of X.

Identifiers
urn:nbn:se:kth:diva-26113 (URN)
Note
QC 20101115Available from: 2010-11-15 Created: 2010-11-15 Last updated: 2010-11-15Bibliographically approved

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