discriminant - O n the fundamental group of the complement...

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On the fundamental group of the complement to a discrlminant variety Igor Dolgachev and Anatoly Libgober 1. Introduction. Let i:V - pn be a closed embedding of a smooth complex algebraic variety into the projective space, ~ c ~ the dual variety of i(V). Its points parametrlze hyperplanes which are tangent to i(V), or equivalently, singular hyperplane sections of In this paper we discuss the group ,i(~-~) and compute it in some special cases. If L ~ pn is a general 2-plane, then by the Zariski-Lefschetz @ v v type theorem ([Z3],[LH]) ~l(Pn-v) - ~I(L-LDV). The intersection L~ is either empty or a plane irreducible curve with nodes and cusps as singularities. Its degree, the number of nodes and cusps can be computed by generalized Pl'ucker formulas (see n°2). It should be sald that the group ~l(P2-C) for a nodal-cuspidal plane curve C is known only in a few cases. We discuss in n°3 the previously known examples of Zarlski ([Zl],[Z2]) of such groups. The presence of cusps is an essential obstacle, since, as it had been recently .i(~2 proven by Fulton-Dellgne (see [D]), -C) is always abellan if C has only nodes. In the above mentioned examples of Zariskl V = pl * The authors were partially supported by the National Science Foundation.
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There is a close relation between the braid groups of Riemann surfaces and the homotopy groups of the diffeomorphisms groups of Riemann surfaces (see [B]). In section 5 we speculate on a possible generalization of this relation in the case of an arbitrary embedding i:V ~n 2. The discriminant variety of a linear system. Let V be a nonsingular projective algebraic variety over complex numbers, L an invertible sheaf on V, E a linear subspace of H0(V,L) , ~(E) the corresponding linear system of divisors on V. Define the discriminant variety Disc(E,L) of the linear system ~(E) as the subset of points x ~ ~(E) such that the corresponding divisor D x is not smooth (every positive divisor is considered as a closed subscheme of V). This set is always closed in the Zariski topology of the projective space and hence has a unique structure of a reduced algebraic subvariety of ~(E). The most interesting case in which we will be involved is the case where ~ is a very ample sheaf and E = HO(v,~). In this case the complete linear system defines a closed embedding i:V - ~(E*). The discrlmlnant variety in this case, denoted simply by Disc(L), coincides with the dual variety i~V) i(V). The latter is defined as the set of all points x in the dual projective space ~(E*~ = ~(E) such that the corresponding hyperplane H x is tangent to i(V) somewhere. An equivalent definition of can be given also as follows (see [KL] p. 335). Z = ~(E*) x iP(E) be the canonical incidence correspondence between points and hyperplanes, pl:Z - ~(E*), p2:Z - ~(E) the projections,
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and i:P I - ~ is the Veronese embedding v n. In this case - Vn(P ) is canonically isomorphic to the space sn(~ l) of all unordered n-tuples of distinct points on ~l The fundamental group of this space is known as the n-th braid group of the Riemann sphere. It has been recomputed by many authors who apparently were not aware of Zariski's papers (see [B]).
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discriminant - O n the fundamental group of the complement...

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