Free and nearly free curves from conic pencils
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.
Abstract
We construct some infinite series of free and nearly free curves using pencils of conics with a base locus of cardinality at most two. These curves have an interesting topology, e.g. a high degree Alexander polynomial that can be explicitly determined, a Milnor fiber homotopy equivalent to a bouquet of circles, or an irreducible translated component in the characteristic variety of their complement. Monodromy eigenspaces in the first cohomology group of the corresponding Milnor fibers are also described in terms of explicit differential forms.
참고문헌 (31)
- M. Amram, "Fundamental groups of tangent conic-line arrangements with singularities up to order 6", Math. Z., 256(4), pp.837~870 (2007)[DOI]
- E. Artal Bartolo, "On fundamental groups of plane curve complements", Ann. Univ. Ferrara Sez. VII Sci. Mat., 61(2), pp.255~262 (2015)[DOI]
- S. Bannai, "Geometry of bisections of elliptic surfaces and Zariski N-plets for conic arrangements", Geom. Dedicata, 178, pp.219~237 (2015)[DOI]
- N. Budur, "Jumping coefficients and spectrum of a hyperplane arrangement", Math. Ann., 347(3), pp.545~579 (2010)[DOI]
- A. Dimca, "Singularities and Topology of Hypersurfaces, Universitext" (1992), Springer-Verlag
- A. Dimca, "Characteristic varieties and constructible sheaves", Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 18(4), pp.365~389 (2007)
- A. Dimca, "Freeness versus maximal global Tjurina number for plane curves", Math. Proc. Cambridge Philos. Soc., 163(1), pp.161~172 (2017)[DOI]
- A. Dimca, "Hyperplane Arrangements, Universitext" (2017), Springer
- A. Dimca, "Curve arrangements, pencils, and Jacobian syzygies", Michigan Math. J., 66(2), pp.347~365 (2017)[DOI]
- A. Dimca, "Koszul complexes and spectra of projective hypersurfaces with isolated singularities", arXiv:1212.1081, ~
- A. Dimca, "On the exponents of free and nearly free projective plane curves", Rev. Mat. Complut., 30(2), pp.259~268 (2017)[DOI]
- A. Dimca, "A computational approach to Milnor fiber cohomology", Forum Math., 29(4), pp.831~846 (2017)
- A. Dimca, "Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves", arXiv: 1609.06818, ~
- A. Dimca, "Computing Milnor fiber monodromy for some projective hypersurfaces", arXiv: 1703.07146, ~
- A. A. du Plessis, "Application of the theory of the discriminant to highly singular plane curves", Math. Proc. Cambridge Philos. Soc., 126(2), pp.259~266 (1999)[DOI]
- M. Falk, "Multinets, resonance varieties, and pencils of plane curves", Compos. Math., 143(4), pp.1069~1088 (2007)[DOI]
- R. V. Gurjar, "Open surfaces with non-positive Euler characteristic", Compositio Math., 99(3), pp.213~229 (1995)
- A. J. de Jong, "Proof of a conjecture of W. Veys", Indag. Math. (N.S.), 6(1), pp.99~104 (1995)[DOI]
- P. Orlik, "Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften, 300" (1992), Springer-Verlag
- A. Ploski, "A bound for the Milnor number of plane curve singularities", Cent. Eur. J. Math., 12(5), pp.688~693 (2014)
- K. Saito, "Theory of logarithmic differential forms and logarithmic vector fields", J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27(2), pp.265~291 (1980)
- M. Saito, "Multiplier ideals, b-function, and spectrum of a hypersurface singularity", Compos. Math., 143(4), pp.1050~1068 (2007)[DOI]
- M. Saito, "Bernstein-Sato polynomials and graded Milnor algebras for projective hyper-surfaces with weighted homogeneous isolated singularities", arXiv:1609.04801, ~
- M. Saito, "Roots of Bernstein-Sato polynomials of homogeneous polynomials with 1-dimensional singular loci", arXiv:1703.05741, ~
- H. Schenck, "Freeness of conic-line arrangements in <TEX>${\mathbb{P}}^2$<", TEX>, 2009, pp.84(2), 235~258
- 신재선, "A bound for the Milnor sum of projective plane curves in terms of GIT", 대한수학회지, 53(2), pp.461~473 (2016), 대한수학회
- A. Simis, "Homology of homogeneous divisors", Israel J. Math., 200(1), pp.449~487 (2014)[DOI]
- H. Terao, "Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula", Invent. Math., 63(1), pp.159~179 (1981)[DOI]
- J. Valles, "Free divisors in a pencil of curves", J. Singul., 11, pp.190~197 (2015)
- W. Veys, "Structure of rational open surfaces with non-positive Euler characteristic", Math. Ann., 312(3), pp.527~548 (1998)[DOI]
- C. T. C. Wall, "Singular Points of Plane Curves, London Mathematical Society Student Texts, 63" (2004), Cambridge University Press