New Asymptotics for the Existence of Plane Curves with Prescribed Singularities

On the Existence of Plane Curves with Prescribed Singularities

An asymptotic existence theorem for plane curves with prescribed singularities

Let d,m1, . . . ,mr be (r + 1) positive integers. Denote by V (d;m1, . . . ,mr) the variety of irreducible (complex) plane curves of degree d having exactly r ordinary singularities of multiplicities m1, . . . ,mr. In most cases, it is still an open problem to know whether this variety is empty or not. In this paper, we will concentrate on the case where the r singularities can be taken in a ge...

On the Exceptional Locus of the Birational Projections of a Normal Surface Singularity into a Plane

Given a normal surface singularity (X,Q) and a birational morphism to a nonsingular surface π : X → S, we investigate the local geometry of the exceptional divisor L of π. We prove that the dimension of the tangent space to L at Q equals the number of exceptional components meeting at Q. Consequences relative to the existence of such birational projections contracting a prescribed number of irr...

Integrability criteria for differential equations on the projective plane

In this article we prove two new criteria for the existence of rational general integral of an algebraic differential equation (cf. [4]) on the complex projective plane. These results are stated in terms of divisors and zero-cycles of a projective variety and are built by using intersection numbers of projective curves and multiplicities of singularities. We also prove a new result giving new s...

On the Dirichlet Problem for Harmonic Maps with Prescribed Singularities

Let (M, g) be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into (M, g) with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps into the hyperbolic plane were constructed. This problem, in the case wh...