On the Singularities of Surfaces Ruled by Conics (original) (raw)

Families of rational surfaces preserving a cusp singularity

Transactions of the American Mathematical Society, 1990

Families of rational surfaces containing resolutions of cusp singularities are explicitly constructed. It is proved that the families constructed are universal deformations at each point. Two different monodromy formulas are established; one of these is shown to be connected to automorphisms of Inoue-Hirzebruch surfaces. Some evidence (but no proof) is offered for the conjecture that finite base changes of the families we construct are the versal-deformation spaces for singular Inoue-Hirzebruch surfaces.

Combinatorics of Rational Surface Singularities

2005

A normal surface singularity is rational if and only if the dual graph of a desingularization satisfies some combinatorial properties. In fact, these graphs are trees. In this paper we give geometric features of these trees. In particular, we prove that the number of vertices of valency > 3 in the dual tree of the minimal desingularization of a rational singularity of multiplicity m > 3 is at most m — 2. MIRAMARE TRIESTE December 1999 E-mail: ledt@gyptis.univ-mrs.fr E-mail: tosun@ictp.trieste.it

Explicit resolutions of double point singularities of surfaces

1999

Locally analytically, any isolated double point occurs as a double cover of a smooth surface. It can be desingularized explicitly via the canonical resolution, as it is very well-known. In this paper we explicitly compute the fundamental cycle of both the canonical and minimal resolution of a double point singularity and we classify those for which the fundamental cycle differs from the fiber cycle. Moreover we compute the conditions that a double point singularity imposes to pluricanonical systems. : 14J17, 32S25.

Computing the singularities of rational surfaces

Mathematics of Computation, 2014

Given a rational projective parametrization P(t, s, v) of a rational projective surface S we present an algorithm such that, with the exception of a finite set (maybe empty) B of projective base points of P, decomposes the projective parameter plane as

Rational conic fibrations of sectional genus two

Advances in Geometry, 2020

Polarized rational surfaces (X, L) of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of 𝔽1 at some points lying on distinct fibers. Ampleness and very ampleness of L are studied in terms of their location. When L is very ample and there is a line contained in X and transverse to the fibers, the conic fibrations (X, L) are classified and a related property concerned with the inflectional locus is discussed.

Superisolated Surface Singularities

2005

In this survey we review part of the theory of superisolated singularities (SIS) of surfaces and its applications including some new and recent developments. The class of SIS singularities is, in some sense, the simplest class of germs of normal surface singularities. Namely, their tangent cones are reduced curves and the geometry and topology of the SIS singularities can be deduced from them. Thus this class contains, in a canonical way, all the complex projective plane curve theory, which gives a series of nice examples and counterexamples. They were introduced by I. Luengo to show the non-smoothness of the µconstant stratum and have been used to answer some other interesting open questions. We review them and the new results on normal surface singularities whose link are rational homology spheres. We also discuss some positive results which have been proved for SIS singularities.

Equisingularity classes of birational projections of normal singularities to a plane

Advances in Mathematics, 2007

Given a birational normal extension O of a two-dimensional local regular ring (R, m), we describe all the equisingularity types of the complete m-primary ideals J in R whose blowing-up X = Bl J (R) has some point Q whose local ring O X,Q is analytically isomorphic to O. * 1 fixed a birational normal extension O of a local regular ring (R, m O ), we describe the equisingularity type of any complete m O -primary ideal J ⊂ R such that its blowing-up X = Bl J (R) has some point Q whose local ring O X,Q is analytically isomorphic to O. In this case, we will say that the surface X contains the singularity O for short, making a slight abuse of language. This is done by describing the Enriques diagram of the cluster of base points of any such ideal J: such a diagram will be called an Enriques diagram for the singularity O. Recall that an Enriques diagram is a tree together with a binary relation (proximity) representing the topological equivalence classes of clusters of points in the plane (see §1.3). Previous works by Spivakovsky and Möhring [12] describe a type of Enriques diagram that exists for any given sandwiched surface singularity (detailed in §2) and provide other types mostly in the case of cyclic quotients (see [12] 2.7) and minimal singularities (see 2.5).

Birational classification of curves on rational surfaces

Nagoya Mathematical Journal, 2010

In this paper we consider the birational classification of pairs (S, L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its Cremona minimal models, i.e. those plane curves which are equivalent to B via a Cremona transformation, and have minimal degree under this condition.

Hypersurface sections and obstructions (rational surface singularities)

Compositio Mathematica, 1991

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On Four-Dimensional Terminal Quotient Singularities

Mathematics of Computation, 1988

We report on an investigation of four-dimensional terminal cyclic quotient singularities which are not Gorenstein. (For simplicity, we focus on quotients by cyclic groups of prime order.) An enumeration, using a computer, of all such singularities for primes < 1600 led us to conjecture a structure theorem for these singularities (which is rather more complicated than the known structure theorem in dimension three). We discuss this conjecture and our evidence for it; we also discuss properties of the anticanonical and antibicanonical linear systems of these singularities. Introduction. The recent successes in understanding the birational geometry of algebraic varieties of dimension greater than two have focused attention on a class of singularities (called terminal singularities) which appear on the birational models which the theory selects. In dimension three, the structure of these terminal singularities is known in some detail: The terminal quotient singularities were classified, in what is now called the "terminal lemma", by several groups of people working independently (cf. [1], [2], [5], [8]), and all other three-dimensional terminal singularities were subsequently classified by work of the first author [6] and of Kollár and Shepherd-Barron [4]. Both of these classifications were based on key results of Reid [10], [11] who reduced the problem to an analysis of quotients of smooth points and double points by cyclic group actions. (A detailed account of the classification can be found in [12].) A consequence of this classification, apparently first observed by Reid [12], is that for any three-dimensional terminal singularity T, the general element of the anticanonical linear system |-KT\ has only canonical singularities. This in turn implies that if we form the double cover of T branched on the general antibicanonical divisor D E [-2Kt\, that double cover will also have only canonical singularities. At first glance, this second property appears to have no advantages over the first, but looking at things in these terms proved decisive in a slightly more global context: Kawamata [3] showed that if T is an "extremal neighborhood" of a rational curve C on a threefold, then the existence of such a divisor D globally on T is sufficient to conclude the existence of a certain kind of birational modification (a "directed flip") centered on C. The first author [7] showed that such divisors always exist, completing the proof of the Minimal Model Theorem for threefolds. (See [7] for more details.