Related papers: On germs of mappings $\mathbb C^2\to\mathbb C^2$

In this article we continue from \cite{sub2-1} the study of normal analytic compactifications of $\cc^2$ from the point of view of their associated pencils of jets of curve germs centered at infinity. If $\bar X$ is a normal analytic…

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key…

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and…

In this paper we study the second fundamental form of the Prym map $P_{g,r}: R_{g,r} \rightarrow {\mathcal A}^{\delta}_{g-1+r}$ in the ramified case $r>0$. We give an expression of it in terms of the second fundamental form of the Torelli…

We give conditions under which a germ of a holomorphic mapping in $\Bbb C^N$, mapping an irreducible real algebraic set into another of the same dimension, is actually algebraic. Let $A\subset \bC^N$ be an irreducible real algebraic set.…

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$…

Let $L/K$ be an extension of number fields that is ramified above $p$. We give a new obstruction to the descent to $K$ of smooth projective varieties defined over $L$. The obstruction is a matrix of $p$-adic numbers that we call ``ramified…

We show that two analytic function germs $(\C^2,0) \to (\C,0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of…

Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X,C)\to (Z,o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X,C)$ contains a…

We compute the homology of the multiple point spaces of stable perturbations of two germs $(\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ of corank 2, one with $n=3$ and one with $n=5$. We use a variety of techniques based on the image computing…

If $R$ is a real analytic set in $\C^n$ (viewed as $\R^{2n}$), then for any point $p\in R$ there is a uniquely defined germ $X_p$ of the smallest complex analytic variety which contains $R_p$, the germ of $R$ at $p$. It is shown that if $R$…

We study holomorphic fixed point germs in two complex variables that are tangent to the identity and have a degenerate characteristic direction. We show that if that characteristic direction is also a characteristic direction for higher…

We describe a method for counting maps of curves of given genus (and variable moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may…

We explicitly calculate some Gromov--Witten correspondences determined by maps of labeled curves of genus zero to the moduli spaces of labeled curves of genus zero. We consider these calculations as the first step towards studying the…

We study correspondences between algebraic curves defined over the separable closure of $\Q$ or $\F_p$.

Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>= 2. Such loci have codimension two. We compute the class…

Let $C$ be a curve of genus 2 and $\psi_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $\phi_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering.…

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we…

The Borel mapping takes germs at $0$ of smooth functions to the sequence of iterated partial derivatives at $0$. We prove that the Borel mapping restricted to the germs of any quasianalytic ultradifferentiable class strictly larger than the…

In this paper we find an explicit formula for the number of topologically different ramified coverings $C\to\CP^1$ (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the…