Intersections of projective varieties and generic projections

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.

Communications in Algebra, 2001

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P n , n ≥ 3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p ≤ n − 1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P n , p ≤ n and the characteristic of the base field is zero. Finally, using our characterization, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu ([4]) on self-linking and complete intersections in positive characteristic.

Publications de l'Institut Math?matique (Belgrade), 2003

We deal with Hirzebruch genera of complete intersections of non-singular projective hyper surfaces. We give the formula for genera of algebraic curve and surfaces and prove that symmetric squares of algebraic curve of genus g > 0 are not projective complete intersections.

arXiv preprint arXiv:0903.3359, 2009

We study ramified covers of the projective plane P 2 . Given a smooth surface S in P n and a generic enough projection P n → P 2 , we get a cover π : 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.

Mathematische Annalen, 1991

Let V be a (n+ 1)-dimensional vector space. Let G=G(r, V) be the Grassmann variety. Let X__ G be a generic complete intersection of type (m 1, m 2 ..... mk) in G. In this paper we study the Hilbert scheme of X. Let Ho be an open irreducible subset of the Hilbert schemes of X which parametrizes smooth irreducible projective subvarieties. Suppose that ~o ~ Ho is the corresponding universal family of subvarieties. Let F: ~o ~ X be the natural map and Y= Im (F). Suppose that Z is a general member of the family Ho. We show that Nz/x | Cz(1) is generated by its sections. Let m = m 1 + m2 +... + mk and mo be the least integer such that H~174 Then we show that Codimx Y>m+mo-n-1. In particular, if m => dim X + n + 1, then every smooth projective subvariety of X is of general type. This gives a generalization of a result of Clemens about curves on generic hypersurfaces in [1]. We also obtained results which shows that the Hilbert scheme of X is smooth at those points corresponding to low degree smooth rational curves. (See 2.7 for the precise statement.)

Mathematische Annalen, 1983

The aim of this note is to use the techniques of reflexive sheaves, , to prove the following theorem:

Inventiones mathematicae, 1991

We study ramified covers of the projective plane. Given a smooth surface S in P^n and a generic enough projection P^n --> P^2, we get a cover $\pi: S \to P^2$, 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.

Journal of the American Mathematical Society, 1991

We study ramified covers of the projective plane P 2 . Given a smooth surface S in P n and a generic enough projection P n → P 2 , we get a cover π : 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.