Immersions associated with holomorphic germs

A holomorphic germ \Phi: (C^2, 0) \to (C^3, 0), singular only at the origin, induces at the links level an immersion of S^3 into S^5. The regular homotopy type of such immersions are determined by their Smale invariant, defined up to a sign ambiguity. In this paper we fix a sign of the Smale invariant and we show that for immersions induced by holomorphic gems the sign-refined Smale invariant is the negative of the number of cross caps appearing in a generic perturbation of \Phi. Using the algebraic method we calculate it for some families of singularities, among others the A-D-E quotient singularities. As a corollary, we obtain that the regular homotopy classes which admit holomorphic representatives are exactly those, which have non-positive sign-refined Smale invariant. This answers a question of Mumford regarding exactly this correspondence. We also determine the sign ambiguity in the topological formulae of Hughes-Melvin and Ekholm-Szucs connecting the Smale invariant with (singular) Seifert surfaces. In the case of holomorphic realizations of Seifert surfaces, we also determine their involved invariants in terms of holomorhic geometry

Coble fourfold, S6S_6-invariant quartic threefolds, and Wiman-Edge sextics

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as $S_6$-representations.Comment: 57 pages; v2: minor changes; v3: referee's comments taken into account; v4: published versio

Classification of SUSY and non-SUSY Chiral Models from Abelian Orbifolds AdS/CFT

We classify compactifications of the type IIB superstring on AdS_{5} x S^{5}/\Gamma, where \Gamma is an abelian group of order n<= 12. Appropriate embedding of \Gamma in the isometry of S^5 yields both SUSY and non-SUSY chiral models that can contain the minimal SUSY standard model or the standard model. New non-SUSY three family models with \Gamma=Z_8 are introduced, which lead to the right Weinberg angle for TeV trinification.Comment: 12 pages, no figur

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European Journal of Mathematics, special issue in memory of W.L. Edg

Simple finite subgroups of the Cremona group of rank 3

We classify all finite simple subgroups in the Cremona group of rank 3Comment: 32 pages, late

Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects

In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European Journal of Mathematics, special issue in memory of W.L. Edg