The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks

Tu, Loring W.

doi:10.1090/S0002-9947-1989-0930069-7

The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks
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Trans. Amer. Math. Soc. 313 (1989), 381-392 Request permission

Abstract:

A degeneracy locus is the set of points where a vector-bundle map has rank at most a given integer. Such a set is symmetric or skew-symmetric according as whether the vector-bundle map is symmetric or skew-symmetric. We prove a connectedness result, first conjectured by Fulton and Lazarsfeld, for skew-symmetric degeneracy loci and for symmetric degeneracy loci of even ranks.

References

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
Spencer Bloch and David Gieseker, The positivity of the Chern classes of an ample vector bundle, Invent. Math. 12 (1971), 112–117. MR 297773, DOI 10.1007/BF01404655
W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math. 146 (1981), no. 3-4, 271–283. MR 611386, DOI 10.1007/BF02392466
William Fulton and Robert Lazarsfeld, Positive polynomials for ample vector bundles, Ann. of Math. (2) 118 (1983), no. 1, 35–60. MR 707160, DOI 10.2307/2006953
Joe Harris and Loring W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), no. 1, 71–84. MR 721453, DOI 10.1016/0040-9383(84)90026-0

The connectedness of symmetric degeneracy loci

odd ranks

Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
Piotr Pragacz, Cycles of isotropic subspaces and formulas for symmetric degeneracy loci, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 189–199. MR 1171269
Andrew John Sommese, Submanifolds of Abelian varieties, Math. Ann. 233 (1978), no. 3, 229–256. MR 466647, DOI 10.1007/BF01405353
Loring W. Tu, Degeneracy loci, Proceedings of the conference on algebraic geometry (Berlin, 1985) Teubner-Texte Math., vol. 92, Teubner, Leipzig, 1986, pp. 296–305. MR 922921
Loring W. Tu, The connectedness of degeneracy loci, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 235–248. MR 1171274