*Dedicated to Micheline Ishay I would like to thank P. Boyland, L. Chierchia, V. Donnay, G. De Martino, C. Gole, J. L. Lebowitz, M. Lyubich, M. Rychlik, I. G. Schwarz, S. Vaienti and especially G. Gallavotti for helpful and enlightening discussions. Particularly warm thanks go to N. Chernov for carefully reading and finding a mistake in an early version; the present paper benefits from several improvements due to his sharp criticism. In addition, I am indebted to P. Collet and, most of all, M. Wojtkowski for introducing me to the magical world of cones. Finally, I thank J. Milnor, director of the Institute for Mathematical Sciences at Stony Brook University, where I was visiting during part of this work, and the Italian C.N.R.-GNFM for providing travel funds.

The mod p cohomology of a space or spectrum X is a module over the mod p Steenrod algebra A. If X is a space, it has additional structure making it an unstable A-algebra, where we recall that an A-module M is unstable if, when p = 2, Sqi x = 0 for all i > IxI and x E M, and, when p > 2, 3ePix = 0 for all 2i + e > JxJ, e = 0 or 1, and x E M. Questions about realizing particular A-modules, algebras, and A-algebras as the cohomology of spaces or spectra are among the most fundamental in algebraic topology. The Hopf invariant one problem [A] and the problem of realizing polynomial algebras [DMW] are two famous examples of ongoing interest. If one regards the first of these as a problem about realizing finite A-modules, and the second of these as a problem about realizing rather large A-modules, it is the purpose of this paper to study the middle ground. We begin with the following conjecture.

One of the themes in algebraic geometry is the study of the relation between the ``topology'' of a smooth projective variety and a (``general'') hyperplane section. Recent results of Nori produce cohomological evidence for a conjecture that a general hypersurface of sufficently large degree should have no ``interesting'' cycles. We compute precise bounds for these results and show by example that there are indeed interesting cycles for degrees that are not high enough. In a different direction Esnault, Nori and Srinivas have shown connectivity for intersections of small multidegree. We show analogous cycle-theoretic connectivity results.