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Complements to Ample Divisors and Singularities

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Handbook of Geometry and Topology of Singularities II

Abstract

The paper reviews recent developments in the study of Alexander invariants of quasi-projective manifolds using methods of singularity theory. Several results in topology of the complements to singular plane curves and hypersurfaces in projective space extended to the case of curves on simply connected smooth projective surfaces.

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Notes

  1. 1.

    I.e. a family of divisors parametrized by \({\mathbb P}^1\).

  2. 2.

    Those are absent in the classical case on pencils of lines \(X={\mathbb P}^2\) of Zariski-van Kampen theorem.

  3. 3.

    We assume that there are no vanishing cycles corresponding to critical points of \(\pi \) and no points of D inside this ball.

  4. 4.

    This implies that \(\sigma _i^{-1}(t_i)=t_it_{i+1}t_i^{-1}, \sigma _i^{-1}(t_{i+1})=t_i\).

  5. 5.

    In this case we call \(Y_{\phi }\) 1-finite.

  6. 6.

    An example of infinite cyclic covers which is infinite in dimension 1 is given by the complement to a set [3] containing 3 points in \({\mathbb P}^1\). Let (a, b) be generators of the free group \(\pi _1({\mathbb P}^1\setminus [3])\) and \(\phi \) is the quotient of the normal subgroup generated by b. Then \({\mathbb P}^1\setminus [3]\) is homotopy equivalent to a wedge of two circles and \(({\mathbb P}^1\setminus [3])_{\phi }\) can be viewed as a real line with the circle attached at each integer point of this line with the covering group \({\mathbb Z}\) acting via translations. In particular \(H_1({\mathbb P}^1\setminus [3])_{\phi },{\mathbb Z})\) is a free abelian group with countably many generators.

  7. 7.

    The condition \(a_0=0\) is equivalent to finite dimensionality of \(H_1(Y_{\phi },{\mathbb Q})\) over \({\mathbb Q}\).

  8. 8.

    I.e. a loop consisting of a path connecting the base point with a point in vicinity of the irreducible component of D, the oriented boundary of a small disk in X transversal to this component of D at its smooth point and not intersecting the other components of D, with the same path used to return back to the base point; orientation of the small disk must be positive i.e. such that its orientation will be compatible with the complex orientations of smooth locus of divisor and the ambient manifold. As an element of the fundamental group, only the conjugacy class of a meridian is well defined.

  9. 9.

    Recall that the ground field here is \({\mathbb C}\). For varieties over non-algebraically closed fields, the inertia group I(x) of \(x \in \Delta \) (which is the subgroup of the decomposition group consisting of automorphisms inducing trivial automorphism of the extension of the residue fields of f(x) by the residue field of x (cf. [104] Expose V, Sect. 2) is a proper subgroup of the decomposition group.

  10. 10.

    Recall that this follows from identification \(H_{2 \dim D}(D,U(1))=H^{2 \dim D}(Y,Y \setminus D,U(1))\) obtained by excision and Lefschetz duality.

  11. 11.

    Recall that Y is simply connected and hence \(\mathcal{L}_{\chi }\) is well defined.

  12. 12.

    The support is assumed to be a reduced variety.

  13. 13.

    However \(Char_1(\pi _1({\mathbb P}^1\setminus [3]),ab)=({\mathbb C}^*)^2\) where [3] is a subset containing 3 points and ab is the abelianization of the free group.

  14. 14.

    \(Spec {\mathbb C}[A]\) is algebraic group with CardTorA connected components with \(({\mathbb C}^*)^{rk A}\) being the component of identity.

  15. 15.

    The order of vanishing of the Fitting ideal of \(H_1(Y_{\phi },{\mathbb C})\) at (1, ...., 1) in general is different than \(rk H_1(Y,{\mathbb C})\) which is the first Betti number of trivial local system.

  16. 16.

    The integer \(d(D^{\chi },\chi )\) is called the depth of the character \(\chi \) of the curve \(D^{\chi }\).

  17. 17.

    We also use that removal 0-dimensional set SingD from a 4-dimensional manifold does not change the first Betti number.

  18. 18.

    We call D a divisor with isolated non-normal crossings, if \(\dim NNC(D)=0\).

  19. 19.

    The convention is that if \(x \notin D\) then D does have normal crossing at x and the dimension of empty set is \(-1\).

  20. 20.

    Only the last claim in (ii) requires singularity of f to be isolated.

  21. 21.

    Which is also the universal cover for \(n \ge 2\) since the fundamental group is abelian in this case.

  22. 22.

    Torsion freeness condition of \(H_1(X,{\mathbb Z})\) is introduced to simplify the exposition.

  23. 23.

    By a polytope we mean a set of solutions to a finite collection of inequalities. All polytopes considered here are bounded (subsets of a unit cube) and hence are the convex hulls of the sets of their vertices. Faces are subsets of the boundary of a polytope which are the convex hulls of a subset of the set of vertices of the polytope. The dimension of a polytope (including a face) is the maximal dimension of the balls in its interior (with the dimension of a vertex being zero).

  24. 24.

    I.e. the set of solutions to a linear inequality.

  25. 25.

    This is a local version of the global construction used in Proposition 10.3.9. Here the rank of the fundamental group coincides with the number of irreducible components of the divisor.

  26. 26.

    The term introduced by A. Nadel in 1990, cf. [169].

  27. 27.

    Strict polytopes of quasiadjunction were described just before Definition 10.4.3.

  28. 28.

    The number of global polytopes of quasi-adjunction is at most \(\prod _{k \in Sing(D)} n(P_k)\) where \(n(P_k)\) is the number of local polytopes of quasi-adjunction at singular point \(P_k\).

  29. 29.

    cf. construction described in Proposition 10.3.9.

  30. 30.

    I.e. not contained properly in a contributing face of the same strict global polytope of quasi-adjunction.

  31. 31.

    Such circle of problems is inspired by conjectural asymptotic of number fields extensions having a given group as the Galois group or the group of its Galois closure, which are unramified outside an arbitrary subset of primes while the size of the norm of discriminant grows [156]: Malle conjectures implies a positive answer to the inverse problem of the Galois with little hope for solution in near future (as is obtaining a characterization of quasi-projective group).

  32. 32.

    The results in this section make this assumption. It should be possible to eliminate it with essential conclusions remaining intact.

  33. 33.

    I.e. the Hesse arrangement of 12 lines formed by lines containing triples of inflection points of plane smooth cubic cf. [137].

  34. 34.

    I.e. such that the codimension of the base locus of the linear system it defines is at least 2.

  35. 35.

    Important results on geometry of such curves were obtained much earlier by italian school, notably B.Segre, Chisini and his school cf. [193].

  36. 36.

    Generic choice assures that the fundamental group of the complement to the intersection with the plane inside this plane is isomorphic to the fundamental group of the complement to the discriminant of the complete linear system. Non-generic section were studies in very special cases. For a recent study cf. [85].

  37. 37.

    I.e. for any \(t\in T\) there is a neighborhood \(U\subset T\) such that \(\Phi ^{-1}(U)\) and \(T\times \Phi ^{-1}(t)\) are equivalent as stratified spaces.

  38. 38.

    The term was coined in [22] in reference to first example found by Zariski in 1930s.

  39. 39.

    Zariski k-tuples are sets of k curves in distinct classes of equivalence (B) but in the same class (D); sometimes, in a more loose usage, the reference is to sets of k curves in distinct classes for some equivalences (A)–(E) but not another.

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Libgober, A. (2021). Complements to Ample Divisors and Singularities. In: Cisneros-Molina, J.L., Lê, D.T., Seade, J. (eds) Handbook of Geometry and Topology of Singularities II. Springer, Cham. https://doi.org/10.1007/978-3-030-78024-1_10

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