Osculatory behavior and second dual varieties of Del Pezzo surfaces (original) (raw)

( de Gruyter 2001 Osculatory behavior and second dual varieties of

2014

Abstract. Let S be a smooth surface embedded in a projective space, whose general osculating space has the expected dimension. Inside the dual variety of S one can consider the second discriminant locus, which parameterizes the hyperplane sections of S having some singular point of multiplicityd 3. In this paper the various components of the second discriminant loci of Del Pezzo surfaces are investigated from a unifying point of view. This allows us to describe the second dual varieties of such surfaces and to understand their singular loci.

A note on degenerations of del Pezzo surfaces

Annales de l’institut Fourier, 2015

We prove that for a Q-Gorenstein degeneration X of del Pezzo surfaces, the number of non-Du Val singularities is at most ρ(X)+ 2. Degenerations with ρ(X) + 2 and ρ(X) + 1 non-Du Val points are investigated.

Del Pezzo surfaces and points in the plane

2021

This thesis will focus on the study of the relationship that exists between Del Pezzo surfaces, a kind of surface defined as "a smooth birationally trivial surface V on which the anticanonical sheaf is ample" and counting points in the projective plane P2(k) built over a finite field k. The result allowing us to to link the two concepts says that a Del Pezzo surface of degree d bigger that 1 is isomorphic to the blowup up of 9-d points in P2(k). Once we have settled the relationship between the two concepts the results will revolve around counting n-tuples of points in P2(k) both from a theoretical and computational point of view. In particular the case of 8-tuples, corresponding to Del Pezzo surfaces of degree 1, will be the one around which most of the work will revolve, culminating with the statement of a degree 8 monic polynomial expressing the number of 888-tuples of points in general position as a function of the dimension of the base field. The work concludes by con...

On the unirationality of del Pezzo surfaces of degree 2

Journal of the London Mathematical Society, 2014

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k. 1 We give conditions to detect rational curves on del Pezzo surfaces of degree two, and thus prove unirationality of these surfaces. For example, we show that if X is a del Pezzo surface of degree two over a field k, and if X contains eight points whose images under the morphism defined by the anticanonical linear system κ : X → P 2 are distinct and avoid the branch locus, then there is a rational curve over k passing through one of these eight points, which implies X is unirational (see Lemma 19). These sufficient conditions, together with some analysis of the Galois representation Gal(k/k) → W (E 7 ) associated to X, and a few auxiliary geometric lemmas, allow us to prove our main result.

The osculatory behavior of surfaces in bfP5{\bf P}^5bfP5

Pacific Journal of Mathematics, 1986

This article focuses on a certain class of surfaces in P\ exploiting the interplay between local projective differential geometry and global algebraic geometry. These are the so-called hypo-osculating surfaces, which have the property that at every point there is a hyperplane which is doubly tangent. The first main result of the paper, obtained by applying £. Cartan's method of moving frames, is that locally any such surface arises as a vector solution of one of the classical partial differential equations of physics (the wave equation or a generalized heat equation). Such equations were studied from a similar vantage point by G. Darboux and C. Segre, among others. The remainder of the paper is concerned with the global classification problem. Standard techniques in singularity theory yield formulas for the inflection cycles, and in the case of ruled surfaces there is an explicit numerical formula. By combining the earlier differential geometric results with Kodaira's classification of surfaces, one is able to arrive at a fairly complete understanding of the inflectionary behavior of hypo-osculating surfaces. In particular, the embedding of P 1 X P 1 as the quartic scroll is conjecturally the unique such smooth surface which is totally uninflected.

The Structure of the Discriminant of Some Space-Curve Singularities

The Quarterly Journal of Mathematics, 2001

Among the space-curve singularities of the simplest type are the so called wedges D = C ∨ L, consisting of a plane-curve singularity C together with a line L transverse to the plane of C. In this note we describe the discriminant of D in terms of C. In particular, we show that the complement of the discriminant of D is a K (π, 1) if the complement of the discriminant of C is a K (π, 1). We also give a formula for the multiplicity of the discriminant of C ∨ L.