About the computation of the signature of surface singularities z^N+g(x,y)=0

In this article we describe our experiences with a parallel SINGULAR-implementation of the signature of a surface singularity defined by z^N+g(x,y)=0.Comment: 6 page

The dimension of the image of the Abel map associated with normal surface singularities

Let (X, o) be a complex normal surface singularity with rational homology sphere link and let Xe be one of its good resolutions. Fix an effective cycle Z supported on the exceptional curve and also a possible Chern class l ′ ∈ H2 (X, e Z). Define Ecal ′ (Z) as the space of effective Cartier divisors on Z and c l ′ (Z) : Ecal ′ (Z) → Picl ′ (Z), the corresponding Abel map. In this note we provide two algorithms, which provide the dimension of the image of the Abel map. Usually, dim Picl ′ (Z) = pg, dim Im(c l ′ (Z)) and codim Im(c l ′ (Z)) are not topological, they are in subtle relationship with cohomologies of certain line bundles. However, we provide combinatorial formulae for them whenever the analytic structure on Xe is generic. The codim Im(c l ′ (Z)) is related with {h 1 (X, e L)}L∈Im(c l ′ (Z)); in order to treat the ‘twisted’ family {h 1 (X, e L0 ⊗ L)}L∈Im(c l ′ (Z)) we need to elaborate a generalization of the Picard group and of the Abel map. The above algorithms are also generalized

The Abel map for surface singularities I. Generalities and examples

Abstract. Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions X → X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l′ ∈ H 2 (X , Z). With these data we define the variety ECal′ (Z ) of those effective Cartier divisors D supported on Z which determine a line bundles OZ (D) with first Chern class l′. Furthermore, we consider the affine space Picl′ (Z) ⊂ H1(OZ∗ ) of isomorphism classes of holomorphic line bundles with Chern class l′ and the Abel map cl′ (Z) : ECal′ (Z) → Picl′ (Z). The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups H1(Z,L), where we might vary the analytic structure (X, o) (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles L ∈ Picl′ (Z). The case of generic line bundles of Picl′ (Z) and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill–Noether theory of reduced smooth projective curves

On the geometry of strongly flat semigroups and their generalizations

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with respect to the Diophantine Frobenius problem, are exactly the numerical semigroups associated with negative de nite Seifert homology spheres via the possible `weights' of the generic $S^1$-orbit. Furthermore, we consider their generalization to the Seifert rational homology sphere case and prove an explicit (up to a Laufer computation sequence) formula for their Frobenius number. The singularities behind are the weighted homogeneous ones, whose several topological and analytical properties are exploited

Surgery formulae for the Seiberg-Witten invariant of plumbed 3-manifolds

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3--manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated with $\mathcal{T}$ and its counting functions, which encode rich topological information. Using the `periodic constant' of the series (with reduced variables) we prove surgery formulae for the normalized Seiberg--Witten invariants: the periodic constant appears as the difference of the Seiberg--Witten invariants associated with $M(\mathcal{T})$ and $M(\mathcal{T}\setminus \mathcal{I})$, where $\mathcal{I}$ is an arbitrary subset of the set of vertices of $\mathcal{T}$

Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the ‘periodic constant’ of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients). We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart–Macdonald–Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic cosntant. Using these dualities we also find a multivariable polynomial generalization of the Seiberg–Witten invariant, and we compute it in terms of lattice points of certain polytopes. All these invariants are also determined via lattice point counting, in this way we establish a completely general topological analogue of formulae of Khovanskii and Morales valid for singularities with non-degenerate Newton principal part

Polar exploration of complex surface germs

We prove that the topological type of a normal surface singularity pX, 0q provides finite bounds for the multiplicity and polar multiplicity of pX, 0q, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of pX, 0q. A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of pX, 0q, which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of pX, 0q through its Nash transform. This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted

Delta invariant of curves on rational surfaces I. An analytic approach

We prove that if (C, 0) is a reduced curve germ on a rational surface singularity (X, 0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair C X. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann-Roch formula, valid for projective normal surfaces, introduced by Blache