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Curves on Heisenberg invariant quartic surfaces in projective 3-space
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(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.

URN: urn:nbn:se:kth:diva-26113OAI: diva2:370144
QC 20101115Available from: 2010-11-15 Created: 2010-11-15 Last updated: 2010-11-15Bibliographically approved
In thesis
1. Topics in computation, numerical methods and algebraic  geometry
Open this publication in new window or tab >>Topics in computation, numerical methods and algebraic  geometry
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.
Trita-MAT. MA, ISSN 1401-2278 ; 10:13
National Category
urn:nbn:se:kth:diva-25941 (URN)978-91-7415-770-3 (ISBN)
Public defence
2010-11-29, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
QC 20101115Available from: 2010-11-15 Created: 2010-11-05 Last updated: 2010-11-15Bibliographically approved

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