Branch Locus Research Articles

Proper holomorphic mappings between rigid polynomial domains in $\mathbb{C}^{n+1}$

We describe the branch locus of proper holomorphic mappings between rigid polynomial domains in $\mathbb{C}^{n+1}$. It appears, in particular, that it is controlled only by the first domain. As an application, we prove that proper holomorphic self-mappings between such domains are biholomorphic.

Asphericity of Moduli Spaces via Curvature

We show that under suitable conditions a branched cover satisfies the same upper curvature bounds as its base space. First we do this when the base space is a metric space satisfying Alexandrov's curvature condition CAT(κ) and the branch locus is complete and convex. Then we treat branched covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. In neither setting do we require that the branching be locally finite. We apply our results to hyperplane complements in several Hermitian symmetric spaces of nonpositive sectional curvature in order to prove that two moduli spaces arising in algebraic geometry are aspherical. These are the moduli spaces of the smooth cubic surfaces in ℂP3 and of the smooth complex Enriques surfaces.

Dihedral coverings of algebraic surfaces and their application

In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following: Let B B be a reduced plane curve of even degree d d having only a a nodes and b b cusps. If 2 a + 6 b > 2 d 2 − 6 d + 6 2a + 6b > 2d^2 - 6d + 6 , then π 1 ( P 2 ∖ B ) \pi _1(\mathbf {P}^2 \setminus B) is non-abelian. Note that Nori’s result implies that π 1 ( P 2 ∖ B ) \pi _1(\mathbf {P}^2 \setminus B) is abelian, provided that 2 a + 6 b > d 2 2a + 6b > d^2 .

Inertia groups and fibers

Let X and Y be proper, normal, connected schemes over a field K, and let f : X → Y be a finite, flat K-morphism which is generically Galois (i.e., the extension of function fields K(Y ) ↪→ K(X) is Galois) with Galois group G. It is well-known that for the Zariski-open complement U ⊆ Y of the branch locus of f , the map f−1(U) → U is a (right) G-torsor. Thus, for any y ∈ U and x ∈ f−1(y), the extension of fields K(x)/K(y) is Galois and the stabilizer in G of x maps isomorphically to the Galois group Gal(K(x)/K(y)). In particular, when the fiber f−1(y) is irreducible, then Gal(K(x)/K(y)) = G. If K is a global field, it is natural to ask how the injection Gal(K(x)/K(y)) ↪→ G relates ‘arithmetic’ inertia groups in Gal(K(x)/K(y)) with ‘geometric’ inertia groups in G, corresponding to ramification in the map f . The same question can be asked more generally when K is the function field of a connected, normal, noetherian scheme S with positive dimension, where ‘arithmetic’ ramification in Gal(K(x)/K(y)) corresponds to ramification in K(x) of the valuations on K(y) arising from codimension 1 points of the normalization of S in K(y). A special case of this question was investigated by S. Beckmann. She considered the case when K is a number field (with integer ring OK), Y = PK , and X is a geometrically connected curve over K. Let a1, . . . , am be the finitely many branch points of f . Since K has characteristic 0, so f is tamely ramified over each ai, the inertia groups of f over the ai’s are cyclic subgroups of G. For any closed point y ∈ PK , it is not difficult to show that the scheme-theoretic closure {y} in P1OK , which is proper over OK , is also quasi-finite and therefore finite over OK . For example, if y, y′ ∈ PK are distinct closed points, then {y} ∩ {y′} is artinian. In particular, when y is a K-rational point distinct from the ai’s, the intersection {y} ∩ {ai} is an artinian closed subscheme of {y} ' Spec(OK). Let Ip(y, ai) ≥ 0 denote the length of the part of {y} ∩ {ai} which lies over p ∈ Spec(OK), so obviously Ip(y, ai) = 0 for all but finitely many p (depending on y and ai). Let Σf denote the finite set of primes p of OK at which one of the following occurs: • some K(ai)/K is ramified at p, • Ip(ai, aj) > 0 for some i 6= j (i.e., the closures {ai} and {aj} in P1OK meet over p), • the p[t]-adic valuation on K(PK) = K(t) is ramified in K(X), • p divides the degree of f . Note that Σf can be effectively determined and depends only on the the geometry of f and the arithmetic in some of the fibers of f . Beckmann proved the following result:

Ample vector bundles and branched coverings

To a degree d covering f : X → Y of projective manifolds there is naturally attached a holomorphic rank (d−1)−vector bundle Ef , just by dualizing f∗(OX) and dividing by the trivial factor. This bundle reflects interesting properties of f or X. It has certain positivity properties: it is nef on the generic curve, as observed in [PS00], the first part to this paper, and even more, it is nef on every curve not contained in the branch locus of f (Lazarsfeld). Often it has a lot of sections ([PS00]). For special Y much more is true: if Y = Pn, then E is ample and spanned (Lazarsfeld); the same being true for some rational homogeneous manifolds (Kim, Manivel) and conjecturally for all (as long as b2 = 1). In the first part to this paper we investigated Fano manifolds and proved nefness in certain cases (del Pezzo manifolds of higher degrees). We suspected that we might have ampleness as well. The first result in this paper is that E is not always ample (2.1): we provide a triple covering over the Fano threefold V5, appearing as a linear section of the Grassmannian G(2, 5) ⊂ P9, for which E is not ample. However the only curves on which E fails to be ample, are lines (not necessarily in the branch locus). Then (2.2) we give a criterion for ampleness, and as a consequence we see that given a covering X → V5, then E is ample if and only if E is ample on every line. Moreover we show that E is always 1−ample. The second part of the paper is devoted to the study of low degree coverings (of Fano manifolds, usually). Again, a result of Lazarsfeld is the starting point: if f : X → Pn has degree d, then the restriction maps

Focal loci of algebraic varieties i

The focal locus ∑x of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x ∈X and a circle with centre O passing through x which osculates X inx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle Nx and the projective ambient space (∈ sends the normal vector O - x to its endpoint O), and in this paper we address two general problems:. 1)Characterize the"degenerate"case where the focal locus is not a hyper surface. 2)Calculate, in the case where ∑x is a hypersurface, its degree (with multiplicity).

Factorization of proper holomorphic mappings through Thullen domains

In this article, we consider a bounded pseudoconvex domain in ${\bf C}^2$ satifying: (a) it admits a proper holomorphic mapping $f$ onto the unit ball $B^2$, and (b) it is simply connected and has a real analytic boundary. According to [Barletta-Bedford, Indiana U. Math. J, 39(1985), 315-338], the strong pseudconvexity of $B^2$ alone yields that such a domain is at the boundary points that are at the same time a smooth point of the branch locus $Z_{df} = \{\det(J_{\bf C} f) = 0\}$. (Notice that [Diederich-Fornaess, Math. Ann., 282 (1988), 681-700] implies that $f$ as well as $Z_{df}$ extends holomorphically across the boundaries.) Our main contribution in this paper is that we have discovered a stronger rigidity (both local and global) in case the target domain is the unit ball. The main results are: THEOREM (Local Rigidity): Let $(M,o)$ be a real analytic normalized weakly spherical pointed CR hypersurface in ${\bf C}^2$ of order $k_0 > 1$. Let $(\Sigma, o)$ be the pointed Siegel hypersurface given by the defining equation $Re w - |z|^2 = 0$. If there is a holomorphic mapping $F:(M,o) \to (\Sigma,o)$ for which $o$ is a regular branch point, then (1) $(M,o)$ is defined by the equation $Re w - |z|^{2k_0} = 0$, and (2) $F(z,w)$ is equivalent to $(z,w) \mapsto (z^{k_0},w)$ up to a composition with elements in $Aut (M,o)$ and $Aut (\Sigma,o)$. THEOREM (Global Rigidity): Let $D$ and $f:D \to B^2$ be as above, and let $f$ be generically $m$-to-1. Assume that its branch locus $Z_{df}$ admits an analytic component $V$ with the following properties: (1) $f$ is locally a $m$-to-1 branched covering with branch locus $V$ at every point of $V \cap \partial D$; (2) $V \cap \partial D$ is connected and contains no singular point of the variety $Z_{df}$. Then $D$ is biholomorphic to $E_m = \{|z|^{2m} + |w|^2 < 1\}$.

The cover associated to a (1,3)-polarized bielliptic abelian surface and its branch locus

Let A be an abelian surface and let |D| be a polarization of type (1,3) on A. If (A,|D|) is not a product of elliptic curves, such a polarization induces a finite morphism Q: A →p2C of degree 6. In this paper we describe the branch locus of Q when A is bielliptic in thesense of K. Hulek and S. H. Weintraub (see [13]), generalizing the results proved by Ch. Birkenhake and H. Lange in [4].

Double coverings of octic arrangements with isolated singularities

In this paper we construct 206 examples of Calabi-Yau manifolds with different Euler numbers. All constructed examples are smooth models of double coverings of $P^3$ branched along an octic surface. We allow 11 types of (not necessary isolated) singularities in the branch locus. Thus we broaden the class of examples studied in math.AG/9902057. For every considered example we compute the Euler number and give a precise description of a resolution of singularities.

Models of Curves and Finite Covers

Let K be a discrete valuation field with ring of integers OK .L etf :X! Y be afi nite morphism of curves overK. In this article, we study some possible relationships between the models over OK ofX and ofY . Three such relationships are listed below. Consider a Galois coverf :X!Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K ,t henX achieves semi-stable reduction over some explicit tame extension of K.B/ .W henK is strictly henselian, we determine the minimal extensionL=K with the property thatXL has semi-stable reduction. Let f :X ! Y be a finite morphism, with g.Y/> 2. We show that if X has a stable model X over OK ,t henY has a stable model Y over OK , and the morphism f extends to a morphism X! Y. Finally, given any finite morphism f :X! Y , is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X! Y ?A s was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ- ations, withf a cyclic cover of any order> 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3. Mathematics Subject Classifications ( 1991): 14G20, 11G20, 14H30, 14H25. Let OK be a Dedekind domain with field of fractions K.L etf :X ! Y be a finite morphism of projective, smooth, and geometrically connected curves over Spec.K/. In this paper, we study some possible relationships between the models of X and of Y. In the first part of the paper, we look at semi-stable and stable models, while in the second part we investigate regular models. Let us describe the content of this paper. Definitions and standard facts about models are reviewed in the first section. Let B Y denote the branch locus of f ,a nd letK.B/ be the compositum, in an algebraic closure of K, of the residue fields of points of B. In the second section, we consider a Galois cover X! Y of degree prime top and show that, ifY has semi-stable reduction over K ,t henX achieves semi-stable reduction over an explicit tame extension of the field K.B/ (Theorem 2.3). When K is strictly Henselian, there exist extensions LX=K and

On proper holomorphic mappings from domains with T-action

AbstractWe describe the branch locus of a proper holomorphic mapping between two smoothly bounded pseudoconvex domains of finite type in under the assumption that the first domain admits a transversal holomorphic action of the unit circle. As an application we show that any proper holomorphic self-mapping of a smoothly bounded pseudoconvex complete circular domain of finite type in is biholomorphic.

Galois covering singularities I

We give a necessary condition for Galois covering singularities to be logterminal or log -canonical singularities, which is also sufficient under acertain restriction on the branch loci of the covering maps. We also give a method constructing explicitly resolutions of 2-dimensional Abel covering singularities.

Sur les fonctions composées ultradifférentiables

Let E be a compact subset of C n and Φ some C n -valued mapping, holomorphic in a neighborhood of E, whose branch locus Σ intersects E. Let g be a C ∞ function which is ∂ -flat on Φ( E). Assume that the jet on E of the composite function g o Φ belongs to some sufficiently regular Carleman class { l!M l }. Then we give conditions which ensure the existence and allow an explicit definition of a sequence M + depending only on M, E and Φ, such that g satisfies optimal Carleman estimates { l!M l + } on Φ( E). The results can be applied to any mapping Φ with homogeneous components.

Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK 2=2p g−1. We prove that forp g≥7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp g=5, 6. Lastly we show that whenp g≥13 all these surfaces are simply-connected.

ON THE FUNDAMENTAL GROUP OF AN ABELIAN COVER

Let X, Y be smooth complex projective varieties of dimension n≥2 and let f: Y→X be a totally ramified abelian cover. Assume that the components of the branch divisor of f are ample. Then the map f*: π1(Y)→π1(X) is surjective and gives rise to a central extension: [Formula: see text] where K is a finite group. Here we show how the kernel K and the cohomology class c(f) ∈ H2(π1(X), K) of (1) can be computed in terms of the Chern classes of the components of the branch divisor of f and of the eigensheaves of [Formula: see text] under the action of the Galois group. Using this result, for any integer m&gt;0, we construct m varieties X1,…, Xm no two of which are homeomorphic, even though they have the same numerical invariants and they are realized as covers of the same projective variety X with the same Galois group, branch locus and inertia subgroups.

On Dihedral Galois Coverings

Abstract In this paper, we shall give a method in constructing dihedral Galois covering with prescribed branch locus. As an application, we shall look into dihedral Galois covering of P2, where torsion elements of the Mordell-Weil group of an elliptic surface play key roles in constructing coverings

Equilibrium triangulations of the complex projective plane

Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of ℂP2 which fits the equilibrium decomposition of ℂP2 in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that ℂP2 modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation ℝP27of the real projective plane. We also determine all tight simplicial embeddings of ℂP210and ℝP27.

Noether's Inequality for Non-complete Algebraic Surfaces of General Type

Let V be a nonsingular projective surface of Kodaira dimension K(V) > 0. Let D be a reduced, effective, nonzero divisor on V with only normal crossings. In the present article, a pair (F, D) is said to be a minimal logarithmic surface of general type, if, by definition, Kv + D is a numerically effective divisor of self intersection number (Kv + D) 2 > 0 and if Kv + D has positive intersection with every exceptional curve of the first kind on V. Here Kv is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka — Yau type inequality (cf ) := (Kv + D) 2 — c2 -by making use of [8; Theorem 5.5]. In the present article, we shall prove that (cf ) > -c2 — 2 provided that the rational map &\Ky+D defined by the complete linear system \KV + D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus pg := h°(V, Kv + D) = -(cf ) + 2 = 3, D is an elliptic curve and V is the canonical resolution in the sense of Horikawa associated with a double covering h: Y -> P. In addition, the branch locus B of h is a reduced curve of degree eight and the singular locus Sing B consists of points of multiplicity < 3 except for at most one simple quadruple point. Introduction This is a succession of the previous paper [9]. We work over an algebraically closed field fe of characteristic zero. Let V be a nonsingular projective surface defined over fe. If V is a minimal surface of general type, we have the following inequality due to M. Noether: This inequality, together with the Noether formula 12%((9V) = c±(V) 2 + c2(V Communicated by K. Saito, May 18, 1990. 1991 Mathematics Subject Classification: 14J29 Department of Mathematics, the National University of Singapore, Singapore.

Proper holomorphic mappings between generalized pseudoellipsoids

We characterize the existence of proper holomorphic mappings between pseudoellipsoids of the form\(\Sigma (\gamma ) = \left\{ {z \in C^n :\sum\limits_{j = 1}^n {\left| {z_j } \right|^{2\gamma j}< 1} } \right\},\gamma = (\gamma _1 ,\gamma _2 ,...,\gamma _n ) \in (R^ + )^n \). If f:Σ(α)→Σ(β) is any such mapping, the existence of a subgroup Γ of Aut (Σ(α)) such that\(f^{ - 1} f(z) = \bigcup\limits_{\psi \in \Gamma } {\left\{ {\psi (z)} \right\}} \) is shown equivalent to a condition on the branch locus of f.

Singular locus of an infinite integral extension

By adjoining the square roots or higher roots of a certain infinite number of polynomials in two variables with rational coefficients, i.e. by constructing suitable infinite abelian coverings of the rational plane, we obtain examples of two-dimensional normal noetherian pseudogeometric surfaces which are locally excellent but which are not globally excellent because they have infinitely many singular points. The branch loci of these coverings are grids, i.e., infinite families of horizontal and vertical lines in the plane. For studying such coverings we establish a normality criterion and discuss the theory of excellent rings. Involved in the construction of these coverings is the total residuation of prime ideals in cofinal infinite families of field extensions and the complete splitting of prime numbers in finite number fields.