Characters of fundamental groups of curve complements and orbifold pencils
The present work is a user’s guide to the results of , where a description of the space of characters of a quasi-projective variety was given in terms of global quotient orbifold pencils.
Below we consider the case of plane curve complements. In particular, an infinite family of curves exhibiting characters of any torsion and depth 3 will be discussed. Also, in the context of line arrangements, it will be shown how geometric tools, such as the existence of orbifold pencils, can replace the group theoretical computations via fundamental groups when studying characters of finite order, specially order two. Finally, we revisit an Alexander-equivalent Zariski pair considered in the literature and show how the existence of such pencils distinguishes both curves.
KeywordsFundamental Group Irreducible Component Characteristic Variety Alexander Polynomial Orbifold Structure
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- A. Adem, J. Leida and Y. Ruan, “Orbifolds and Stringy Topology”, Cambridge University Press. 2007.Google Scholar
- E. Artal, Sur les couples de Zariski, J. Algebraic Geom. 4 (1994), 223–247.Google Scholar
- E. Artal, J. Carmona, J. I. Cogolludo and M.Á. Marco, Invariants of combinatorial line arrangements and Rybnikov’s example, Singularity theory and its applications, Adv. Stud. Pure Math., vol. 43, Math. Soc. Japan, Tokyo, 2006, pp. 1–34.Google Scholar
- E. Artal, J. I. Cogolludo-Agustín and A. Libgober, Depth of cohomology support loci for quasi-projective varieties via orbifold pencils, J. Reine Angew. Math., to appear, also available at arXiv:1008.2018 [math.AG].Google Scholar
- E. Artal, J. I. Cogolludo-Agustín and D. Matei, Characteristic varieties of quasi-projective manifolds and orbifolds, Preprint available at arXiv:1005.4761v2 [math.AG], 2010.Google Scholar
- E. Artal, J. I. Cogolludo and H.O. Tokunaga, A survey on Zariski pairs, Algebraic geometry in East Asia—Hanoi 2005, Adv. Stud. Pure Math., vol. 50, Math. Soc. Japan, Tokyo, 2008, pp. 1–100.Google Scholar
- J. I. Cogolludo-Agustín and A. Libgober, Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves, Preprint available at arXiv:1008.2018v2 [math.AG], 2010.Google Scholar
- J. I. Cogolludo-Agustín and M.Á. Marco Buzunáriz, The Max Noether fundamental theorem is combinatorial, Preprint available at arXiv:1002.2325v1 [math.AG], 2009.Google Scholar
- A. Dimca, Pencils of plane curves and characteristic varieties Preprint available at math.AG/0606442, 2006.Google Scholar
- A. Dimca, S. Papadima and A.I. Suciu, Formality, Alexander invariants, and a question of Serre, Preprint available at arXiv:math/0512480v3 [math.AT], 2005.Google Scholar
- D. Eisenbud, “Commutative Algebra. With a View Toward Algebraic Geometry”, Graduate Texts in Mathematics, Vol. 150. Springer-Verlag, New York, 1995.Google Scholar
- R. Friedman and J. W. Morgan, “Smooth Four-manifolds and Complex Surfaces”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994.Google Scholar
- A. Libgober and S. Yuzvinsky, Cohomology of local systems, Arrangements—Tokyo 1998, Kinokuniya, Tokyo, 2000, pp. 169–184.Google Scholar
- A.I. Suciu, Fundamental groups of line arrangements: enumerative aspects, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., 2001, pp. 43–79.Google Scholar
- O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. of Math. 51 (1929).Google Scholar