Audun Holme

The geometric and numerical properties of duality in projective algebraic geometry

In this paper we investigate some fundamental geometric and numerical properties ofduality for projective varieties inPkN=PN. We take a point of view which in our opinion is somewhat moregeometric and lessalgebraic andnumerical than what has been customary in the literature, and find that this can some times yield simpler and more natural proofs, as well as yield additional insight into the situation. We first recall the standard definitions of thedual variety and theconormal scheme, introducing classical numerical invariants associated with duality. In section 2 we recall the well known duality properties these invariants have, and which was noted first byT. Urabe. In section 3 we investigate the connection between these invariants andChern classes in the singular case. In section 4 we give a treatment of the dual variety of a hyperplane section of X, and the dual procedure of taking the dual of a projection of X. This simplifies the proofs of some very interesting theorems due toR. Piene. Section 5 contains a new and simpler proof of a theorem ofA. Hefez and S. L. Kleiman. Section 6 contains some further results, geometric in nature.

Codimension 2 subvarieties of projective space

The motivation for the present paper is theHartshorne Conjecture on complete intersections inPn, forn≥6, and in the codimension 2 case: Any smooth codimension 2 subvarietyX ofPn is conjectured to be a complete intersection forn≥6. We prove this conjecture for all varieties with degree below a certain bound, which represents an improvement of the numerical information available untill now.

A combinatorial proof of the duality defect conjecture in codimension 2

In 1974 Robin Hartshorne published the survey article [8]. This article has been very in/uential. In it he makes, among others, the following conjecture: Suppose that X is a smooth, irreducible subvariety of PN , of dimension n¿ 2N=3. Is it then true that X is always a (scheme theoretic) complete intersection? For the time being one may, to 6x ideas, assume the ground 6eld to be C. But in fact all results in this paper remain valid over any algebraically closed 6eld, with the obvious modi6cations if the 6eld is not algebraically closed. The 6rst case to which the conjecture applies is 5-folds in P7, but as no counterexample is known even for 4-folds in P6, this case is usually also included in the conjecture. For P5 a smooth, codimension 2 subvariety, not a complete intersection, is provided by P1 × P2 as embedded into P5 by the Segre-embedding. As this subvariety is of degree 3 and not contained in a hyperplane, it is not a complete intersection. For P3 and P4 there is an abundance of such subvarieties. In the codimension 3 case, a smooth subvariety just outside the area to which the conjecture applies, is the Grassmanian G(1; 4) of lines in P4, as embedded into P9 by the Pl< ucker embedding. This subvariety is also not contained in a hyperplane, and as it is of degree 5, it is thus not a complete intersection. Of course, in the codimension 1 case the assertion is valid in complete generality and essentially amounts to the Hauptidealsatz, the Principal Ideal Theorem. Hartshorne’s Complete Intersection Conjecture can therefore be understood as a vast, conjectural, geometric generalization of this important algebraic theorem. The conjecture is still completely open today, 25 years after Hartshorne committed it to the printed page. Some initial progress was made when Zak, in [21], proved the weaker conjecture of Linear Normality, also posed by Hartshorne [8]. Also, Schneider

Higher Geometry in the Projective Plane

We now replace the field of real numbers ℝ by a general field k. All previous constructions and definitions carry over to general fields with obvious modifications, and we start with the formal definition of an affine or projective (plane) algebraic curve over a field k. Likewise formal definitions of affine restriction and projective closure of such curves are given, and the interplay between these concepts is explored, as well as smooth and singular point on them. The properties of intersection between a line and an affine or projective curve is examined and the tangent star of a curve et a point is defined. The concepts of projective equivalence and asymptotes are introduced, and the class of general conchoids is defined, an important example being the Conchoid of Nicomedes. The dual curve is defined, this being merely the top of a mighty iceberg, to be explored at a later stage.

More on Duality

This final chapter is devoted to duality, the dual variety and the conormal scheme of an embedded projective variety are given as applications. Reflexivity and biduality are studied, in particular duality of hyperplane sections and projections. An application we present here is a very nice theorem of Hefez and Kleiman. Finally we give a brief presentation of some further results on duality and reflexivity.

Conormal Sheaf and Projective Bundles

This chapter starts with introducing the conormal sheaf, then proceeds to the sheaf of Kahler differentials of an S-scheme X. Then comes the construction of the universal 1-quotient on Open image in new window , leading up to the Segre embedding for projective bundles. The chapter concludes with base extensions of projective morphisms and the concept of regular schemes, also the concept of a formally etale morphism is defined.

Constructions and Representable Functors

This chapter proceeds to treat products and coproducts in the usual framework of representable functors.

Cohomology Theory on Schemes

This chapter opens with a short reminder of some homological algebra, used to treat derived functors and Grothendieck cohomology. This is complemented by the Cech cohomology for abelian sheaves on a topological space.

The Riemann-Roch Theorem

This chapter is on the Riemann-Roch Theorem. We start with Hirzebruch’s Riemann-Roch Theorem, and deduce from it the Riemann-Roch Theorem for curves, and for surfaces. We state the general Grothendieck’s Riemann-Roch theorem, deducing that of Hirzebruch from it.

Characteristic Classes in Algebraic Geometry

This chapter is on characteristic classes. We first explain some basic facts on Open image in new window and then proceed to Chern classes, Chern characters and Todd classes. After comments on the singular case we define homological Segre classes and proceed to a study of the Grothendieck Group K(X). Apart from these comments on the singular case, we assume that all schemes are non singular for the remainder of this book. But using our references, the task of extending the treatment which follows to cover the singular case should be straightforward.