Related papers: Singular locus of rational ruled surfaces

We investigate the existence, and lack of unicity, of a holomorphic fibration by discs transversal to a rational curve in a complex surface.

We determine the image of the monodromy map for meromorphic projective structures with poles of orders greater than two. This proves the analogue of a theorem of Gallo-Kapovich-Marden, and answers a question of Allegretti and Bridgeland.…

We prove that every irreducible component of semi-regular loci of effective line bundles in the Picard scheme of a smooth projective variety has at worst rational singularities. This generalizes Kempf's result on rational singularities of…

We investigate geometric invariants of cuspidal edges on focal surfaces of regular surface. In particular, we shall clarify the sign of the singular curvature at a cuspidal edge on a focal surface using singularities of parallel surface of…

We prove that the Euler-Chow series for ruled surfaces and scrolls is rational by means of an explicit computation.

A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal…

We prove a generalisation of Rudin's theorem on proper holomorphic maps from the unit ball to the case of proper holomorphic maps from pseudoellipsoids.

We prove existence of reflexive sheaves on singular surfaces and threefolds with prescribed numerical invariants and study their moduli.

Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms…

We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.

Let $p \geqslant 5$ be a prime number. In this article we provide a complete and explicit description of the locus formed by the compact Riemann surfaces of genus $2(p-1)$ that are endowed with a group of automorphisms of order $4p$. In…

The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we give the classification of regular distributions on rational…

The intersection matrix of a finite simplicial complex has as each of its entries the rank of the intersection of its respective simplices. We prove that such matrix defines the triangulation of a closed connected surface up to isomorphism.

We classify $G$-solid rational surfaces over the field of complex numbers.

In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We show that associating to a Morin map its singular strata defines a ring homomorphism to…

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…

A rational map $\phi: \mathbb{P}^1 \to \mathbb{P}^1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat^{tm}_2$ can be parametrized by…

A meromorphic quadratic differential on a compact Riemann surface defines a complex projective structure away from the poles via the Schwarzian equation. In this article we first prove the analogue of Thurston's Grafting Theorem for the…

If (X,0) is a complex surface germ with a non-isolated singular locus we describe its singular link L of (X,0) and we show that the topology of L determines the topology of the normalization.

The purpose of this note is to show that the regular locus of a complex variety is locally parabolic at the singular set. This yields that the regular locus of a compact complex variety, e.g., of a projective variety, is parabolic. We give…