A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers and let be the set of all integral, projective and nondegenerate varieties of degree and dimension in the projective space , such that, for all , does not lie on any variety of dimension and degree . We say that a variety satisfies a flag condition of type if belongs to . In this paper, under the hypotheses , we determine an upper bound , depending only on , for the number , where denotes the geometric genus of . In case and , the study of an upper bound for the geometric genus has a quite long history and, for , and , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension in as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in . Next we discuss how far is from and show a sort of lifting theorem which states that, at least in certain cases, the varieties of maximal geometric genus must in fact lie on a flag such as , where denotes a subvariety of of degree and dimension . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.

     

A tight bound on the projective dimension of four quadrics

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.     

A Positivstellensatz for projective real varieties

Given two positive definite forms ${f,\,g\in\mathbb {R}[x_0,\ldots,x_n]}$ , we prove that fg ^N is a sum of squares of forms for all sufficiently large N?≥?0. We generalize this result to projective ${\mathbb {R}}$ -varieties X as follows. Suppose that X is reduced without one-dimensional irreducible components, and ${X(\mathbb {R})}$ is Zariski dense in X. Given everywhere positive global sections f of ${L^{\otimes2}}$ and g of ${M^{\otimes2}}$ , where L, M are invertible sheaves on X and M is ample, fg ^N is a sum of squares of sections of ${L\otimes M^ {\otimes N}}$ for all large N?≥?0. In fact we prove a much more general version with semi-algebraic constraints, defined by sections of invertible sheaves. For nonsingular curves and surfaces and sufficiently regular constraints, the result remains true even if f is just nonnegative. The main tools are local-global principles for sums of squares, and on the other hand an existence theorem for totally real global sections of invertible sheaves, which is the second main result of this paper. For this theorem, X may be quasi-projective, but again should not have curve components. In fact, this result is false for curves in general.     

Rigid geometry on projective varieties

We prove rigidity of various types of holomorphic geometric structures on smooth complex projective varieties.     

Vafa-Witten bound on the complex projective space

We prove that the largest first eigenvalue of the Dirac operator among all Hermitian metrics on the complex projective space of odd dimension m, larger than the Fubini-Study metric is bounded by (2m(m+1))^1/2. Mathematics Subject Classification (2000): 53C27, 58J50, 58J60.     

Lines on projective varieties and applications

The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined by quadratic equations the base locus of the projective second fundamental form at a general point coincides, as a scheme, with the variety of lines. The second part concerns the problem of extending embedded projective manifolds, using the geometry of the variety of lines. Some applications to the case of homogeneous manifolds are included.     

Simple connectedness of projective varieties

A Lefschetz type theorem is proven relating the algebraic fundamental group of a smooth projective variety to the algebraic fundamental group of a subvariety set theoretically defined by forms.

     

Segre classes on smooth projective toric varieties

We provide a generalization of the algorithm of Eklund, Jost and Peterson for computing Segre classes of closed subschemes of projective $k$ -space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.