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ALGORITHMS FOR SINGULARITIES AND REAL STRUCTURES OF WEAK DEL PEZZO SURFACES

NIELS LUBBES

King Abdullah University of Science and Technology, Thuwal, Kingdom of Saudi Arabia

https://doi.org/10.1142/S0219498813501582Cited by:0 (Source: Crossref)

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.

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References

R. Blum, Geometry and Differential Geometry, Lecture Notes in Mathematics 792 (Springer, Berlin, 1980) pp. 213–221. Crossref, Google Scholar
N. Bourbaki , Lie Groups and Lie Algebras , Elements of Mathematics ( Springer-Verlag , Berlin , 2002 ) . Crossref, Google Scholar
W. A. de Graaf , Lie Algebras: Theory and Algorithms , North-Holland Mathematical Library 56 ( North-Holland Publishing , Amsterdam , 2000 ) . Google Scholar
I. Dolgachev, Classical algebraic geometry: A modern view (2012), ISBN 1-107-01765-3, URL http://www.math.lsa.umich.edu/~idolga/CAG.pdf . Google Scholar
P. Du Val, Proc. Camb. Philos. Soc. 30, 453 (1934), DOI: 10.1017/S030500410001269X. Crossref, Web of Science, Google Scholar
P. Griffiths and J. Harris , Principles of Algebraic Geometry , Pure and Applied Mathematics ( John Wiley , New York , 1978 ) . Google Scholar
R. Lazarsfeld, Positivity in Algebraic Geometry. I: Classical Setting: Line Bundles and Linear Series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Vol. 48 (Springer-Verlag, Berlin, 2004) . Google Scholar
N. Lubbes, Minimal families of curves on surfaces, Ph.D thesis (2011) 1–274 . Google Scholar
N. Lubbes, Minimal families of curves on surfaces, preprint (2013), [math.AG] , arXiv:1302.6687 . Google Scholar
N. Lubbes, Families of circles on surfaces, preprint (2013), [math.AG] , arXiv:1302.6710 . Google Scholar
N. Lubbes and J. Schicho, J. Algebraic Combin. 1 (2010). Google Scholar
Ju. I. Manin, Inst. Hautes Études Sci. Publ. Math. 30, 55 (1966). Crossref, Google Scholar
K. Matsuki , Introduction to the Mori Program , Universitext ( Springer-Verlag , New York , 2002 ) . Crossref, Google Scholar
T. Oshima, Classification of subsystems of a root system (2006), http://arxiv.org/abs/math/0611904 . Google Scholar
H. Pottmann, L. Shi and M. Skopenkov, Comput. Aided Geom. Design 29(1), 77 (2012), DOI: 10.1016/j.cagd.2011.10.002. Crossref, Web of Science, Google Scholar
J. Schicho, Beitr. Alg. Geom. 42, 71 (2001). Web of Science, Google Scholar
R. Silhol , Real Algebraic Surfaces , Lecture Notes in Mathematics 1392 ( Springer-Verlag , Berlin , 1989 ) . Crossref, Google Scholar
W. A. Stein et al., Sage mathematics software, The Sage Development Team (2012), http://www.sagemath.org . Google Scholar
N. Takeuchi, Proc. Amer. Math. Soc. 100(1), 145 (1987). Web of Science, Google Scholar
N. Takeuchi, Hokkaido Math. J. 29(1), 119 (2000), DOI: 10.14492/hokmj/1350912960. Crossref, Google Scholar
C. T. C. Wall, J. Reine Angew. Math. 1987(376), 47 (1987). Google Scholar