On Veronesean surfaces (original) (raw)

Canadian Journal of Mathematics, 1993

We consider the blowing up of ℙ2 at s sufficiently general distinct points and its projective embedding by the linear system of the curves of a given degree through the points. We study the ideal of the resulting (Veronesean) surface and find that it can be described by two matrices of linear forms, in the sense that it is generated by the entries of the product matrix and the minors of complementary orders of the two matrices. By cutting the surface twice with general hyperplanes, we also obtain some information about the generation (or even the resolution) of certain classes of points in projective space.

Resolution of Veronese Embedding of plane curves

2010

Let C be a smooth (irreducible) curve of degree d in P 2 . Let P 2 ֒→ P 5 be the Veronese embedding and let I C denote the homogeneous ideal of C on P 5 . In this note we explicitly write down the minimal free resolution of I C for d ≥ 2.

GEOMETRY OF FAMILIES OF NODAL CURVES ON THE BLOWN-UP PROJECTIVE PLANE

1997

Let P2r be the projective plane blown up at r generic points. Denote by E0, E1, . . . , Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on P2r respectively. We consider the variety Virr(d; d1, . . . , dr; k) of all irreducible curves C in |dE0 P r i=1 diEi| with k nodes as the only singularities and give asymptoti- cally nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptyness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r � 9 we give the com- plete answer concerning the existence of nodal curves in Virr(d; d1, . . . , dr; k).

On multiples of divisors associated to Veronese embeddings with defective secant variety

Bulletin of the Belgian Mathematical Society - Simon Stevin

In this note we consider multiples aD, where D is a divisor of the blow-up of P n along points in general position which appears in the Alexander and Hirschowitz list of Veronese embeddings having defective secant varieties. In particular we show that there is such a D with h 1 (X, D) > 0 and h 1 (X, 2D) = 0.

GENERAL BLOW-UPS OF THE PROJECTIVE PLANE

We study linear series on a projective plane blown up in a bunch of general points. Such series arise from plane curves of xed degree with assigned fat base points. We give conditions (expressed as inequalities involving the number of points and the degree of the plane curves) on these series to be base point free, i.e. to dene a morphism to a projective space. We also provide conditions for the morphism to be a higher order embedding. In the discussion of the optimality of obtained results we relate them to the Nagata Conjecture expressed in the language of Seshadri constants and we give a lower bound on these invariants.

On ramified covers of the projective plane 1 (with appendix by E. Shustin)

We study ramified covers of the projective plane. Given a smooth surface S in P^n and a generic enough projection, we get a cover of the projective plane f: S --> P^2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise: First, What is the geography of branch curves among all cuspidal-nodal curves? And second, what is the geometry of branch curves; i.e., how can one distinguish a branch curve from a non branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e., form a special 0-cycle on the plane. We start with reviewing what is known about the answers to these questions, both simple and some non-trivial results. Secondly, the classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in P^3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. We also review examples of small degree. In addition, the Appendix written by E. Shustin shows the existence of new Zariski pairs.

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).