Mathematische Zeitschrift

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology of a line arrangement which generalizes the $$\mathcal {I}$$ -invariant introduced by Artal, Florens and the author. This new invariant is called the loop-linking number in the present paper. We refine the result of Cadegan-Schlieper by proving that the loop-linking number is an invariant of the homeomorphism type of the arrangement complement. We give two effective methods to compute this invariant, both are based on the braid monodromy. As an application, we detect an arithmetic Zariski pair of arrangements with 11 lines whose coefficients are in the 5th cyclotomic field. Furthermore, we also prove that the fundamental groups of their complements are not isomorphic; it is the Zariski pair with the fewest number of lines which have this property. We also detect an arithmetic Zariski triple with 12 lines whose complements have non-isomorphic fundamental groups. In the appendix, we give 29 combinatorial types which lead to similar ordered arithmetic Zariski pairs of 11 lines. To conclude this paper, we give a additivity theorem for the union of arrangements. This first allows us to prove that the complements of Rybnikov’s arrangements are not homeomorphic, and then leads us to a generalization of Rybnikov’s result. Lastly, we use it to prove the existence of homotopy-equivalent lattice-isomorphic arrangements which have non-homeomorphic complements.

We introduce bilinear forms in a flag in a complete intersection local $$\mathbb {R}$$ -algebra of dimension 0, related to the Eisenbud–Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable with finite dimensional commutative algebra methods.

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface $$S=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: z_1^5+z_2^3+z_3^2=0\}$$ with the 5-dimensional sphere $${\mathbb {S}}_{\varepsilon }^5=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: \vert z_1\vert ^2+\vert z_2\vert ^2+\vert z_3\vert ^2=\varepsilon ^2\}$$. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S, 0) with the sphere $${\mathbb S}^5_\varepsilon $$ of radius $$\varepsilon $$ small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve in (S, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding $$E_8$$-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor $$(1-t)$$ for an irreducible one.) We show that, under conditions similar to those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two rational parameters describing allowable singularities. For the extreme values of the parameters, we obtain the stacks of stable limits of $$A_n$$ and $$D_n$$ singularities, and the quotients of the miniversal deformation spaces of these singularities by natural $$\mathbb G _m$$ -actions. We interpret the intermediate spaces as log canonical models of the stacks of stable limits of $$A_n$$ and $$D_n$$ singularities.

To prove Fourier restriction estimate using polynomial partitioning, Guth introduced the concept of k-broad part of regular $$L^p$$ norm and obtained sharp k-broad restriction estimates. To go from k-broad estimates to regular $$L^p$$ estimates, Guth employed $$l^2$$ decoupling result. In this article, similar to the technique introduced by Bourgain-Guth, we establish an analogue to go from regular $$L^p$$ norm to its $$(m+1)$$ -broad part, as the error terms we have the restricted k-broad parts ( $$k=2,\ldots ,m$$ ). To analyze the restricted k-broadness, we prove an $$l^p$$ decoupling result, which can be applied to handle the error terms and recover Guth’s linear restriction estimates.