Characteristic varieties of hypersurface complements

Multivariable Alexander Invariants of Hypersurface Complements

We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin’s vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degen...

L–betti Numbers of Hypersurface Complements

In [DJL07] it was shown that if A is an affine hyperplane arrangement in Cn, then at most one of the L2–Betti numbers b i (C n \ A, id) is non–zero. In this note we prove an analogous statement for complements of complex affine hypersurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of [FLM09, LM06] about L2–Betti numbers of pl...

On the Alexander invariants of hypersurface complements

Massey Products of Complex Hypersurface Complements

The study of the topology of hypersurface complements is a classical subject in algebraic geometry. Most of what is known about these spaces is related to invariants of their rational homotopy type. In this paper, we attempt to show that their Fp-homotopy type captures in general more information than the Q-homotopy type, where Fp is the prime field of p elements. Let X be the complement to a h...

Hypersurface Singularities in Positive Characteristic

We present results on multiplicity theory. Differential operators on smooth schemes play a central role in the study of the multiplicity of an embedded hypersurface at a point. This follows from the fact that the multiplicity is defined by the Taylor development of the defining equation at such point. On the other hand, the multiplicity of a hypersurface at a point can be expressed in terms of ...