The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let $$f:S'\longrightarrow S$$ f : S ′ ⟶ S be a cyclic branched covering of smooth projective surfaces over $${\mathbb {C}}$$ C whose branch locus $$\Delta \subset S$$ Δ ⊂ S is a smooth ample divisor. Pick a very ample complete linear system $$|{\mathcal {H}}|$$ | H | on S, such that the polarized surface $$(S, |{\mathcal {H}}|)$$ ( S , | H | ) is not a scroll nor has rational hyperplane sections. For the general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | consider the $$\mu _{n}$$ μ n -equivariant isogeny decomposition of the Prym variety $${{\,\mathrm{Prym}\,}}(C'/C)$$ Prym ( C ′ / C ) of the induced covering $$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$ f : C ′ : = f - 1 ( C ) ⟶ C : $$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$ Prym ( C ′ / C ) ∼ ∏ d | n , d ≠ 1 P d ( C ′ / C ) . We show that for the very general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | the isogeny component $${\mathcal {P}}_{d}(C'/C)$$ P d ( C ′ / C ) is $$\mu _{d}$$ μ d -simple with $${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$ End μ d ( P d ( C ′ / C ) ) ≅ Z [ ζ d ] . In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$ P d ( C ′ / C ) ⊂ Jac ( C ′ ) ⟶ Alb ( S ′ ) .

Double cover K3 surfaces of Hirzebruch surfaces

General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

Limit points of the branch locus of 𝓜g

Let 𝓜g be the moduli space of compact connected hyperbolic surfaces of genus g ≥ 2, and 𝓑g ⊂ 𝓜g its branch locus. Let $\begin{array}{} \widehat{{\mathcal{M}}_{g}} \end{array} $ be the Deligne–Mumford compactification of the moduli space of smooth, complete, connected surfaces of genus g ≥ 2 over ℂ. The branch locus 𝓑g is stratified by smooth locally closed equisymmetric strata, where a stratum consists of hyperbolic surfaces with equivalent action of their orientation-preserving isometry group. Any stratum can be determined by a certain epimorphism Φ. In this paper, for any of these strata, we describe the topological type of its limits points in 𝓜͡g in terms of Φ. We apply our method to the 2-complex dimensional stratum corresponding to the pyramidal hyperbolic surfaces.

Pruned Double Hurwitz Numbers

Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and $\infty$ and only simple ramification else. These objects feature interesting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in Do and Norbury. We show that pruned double Hurwitz numbers, similar to usual double Hurwitz numbers, satisfy a cut-and-join recursion and are piecewise polynomial with respect to the entries of the two special ramification profiles. Furthermore, double Hurwitz numbers can be computed from pruned double Hurwitz numbers. To sum up, it can be said that pruned double Hurwitz numbers count a relevant subset of covers, leading to considerably smaller numbers and computations, but still featuring the important properties we can observe for double Hurwitz numbers.

On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus [Formula: see text] with one boundary component is connected and in the case of non-orientable Klein surfaces it has [Formula: see text] components, if [Formula: see text] is odd, and [Formula: see text] components for even [Formula: see text]. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

Milnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

On the traceless SU(2) character variety of the 6-punctured 2-sphere

We exhibit the traceless SU(2) character variety of a 6-punctured 2-sphere as a 2-fold branched cover of [Formula: see text], branched over the singular Kummer surface, with the branch locus in [Formula: see text] corresponding to the binary dihedral representations. This follows from an analysis of the map induced on SU(2) character varieties by the 2-fold branched cover [Formula: see text] branched over [Formula: see text] points, combined with the theorem of Narasimhan–Ramanan which identifies [Formula: see text] with [Formula: see text]. The singular points of [Formula: see text] correspond to abelian representations, and we prove that each has a neighborhood in [Formula: see text] homeomorphic to a cone on [Formula: see text].