Algorithms for singularities and real structures of weak Del Pezzo surfaces

Type
Article

Authors
Lubbes, Niels

KAUST Department
Computer Science Program
Visual Computing Center (VCC)

Preprint Posting Date
2013-02-27

Date
2014-02-25

Abstract
In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc.30 (1934) 453–491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math.1987(375/376) (1987) 47–66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is well-known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.

Citation
LUBBES, N. (2014). ALGORITHMS FOR SINGULARITIES AND REAL STRUCTURES OF WEAK DEL PEZZO SURFACES. Journal of Algebra and Its Applications, 13(05), 1350158. doi:10.1142/s0219498813501582

Acknowledgements
It is my pleasure to acknowledge that the many computations with Josef Schicho is a major contribution to this paper. Also he recognized the Pappus configuration of Example 7. I would like to thank Michael Harrison for useful discussions concerning root subsystems. I would like to thank Ulrich Derenthal for informing me of a mistake in a previous version of this paper. The algorithms were implemented using the computer algebra system Sage ([18]). This research was supported by the Austrian Science Fund (FWF): project P21461.

Publisher
World Scientific Pub Co Pte Lt

Journal
Journal of Algebra and Its Applications

DOI
10.1142/S0219498813501582

arXiv
1302.6678

Additional Links
https://www.worldscientific.com/doi/abs/10.1142/S0219498813501582http://arxiv.org/pdf/1302.6678

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