Surfaces of Degree 10 in the Projective Fourspace via Linear Systems and Linkage

The paper discusses the classification of surfaces of degree 10 and sectional genus 9 and 10. The surfaces of degree at most 9 are described through classical work dating from the last century up to recent years, while surfaces of degree 10 and other sectional genera are studied elsewhere. We use relations between multisecants, linear systems, syzygies and linkage to describe the geometry of each surface. We want in fact to stress the importance of multisecants and syzygies for the study of these surfaces. Adjunction, which provided efficient arguments for the classification of surfaces of smaller degrees, here appears to be less effective and will play almost no role in the proofs. We show that there are 8 different families of smooth surfaces of degree 10 and sectional genus 9 and 10. The families are determined by numerical data such as the sectional genus, the Euler characteristic, the number of 6-secants to the surface and the number of 5-secants to the surface which meet a general plane. For each type we describe the linear system giving the embedding in P^4, the resolution of the ideal, the geometry of the surface in terms of curves on the surface and hypersurfaces containing the surface, and the liaison class; in particular we find minimal elements in the even liaison class. Each type corresponds to an irreducible, unirational component of the Hilbert scheme, and the dimension is computed.Comment: 52 pages, plain Te

The moduli space of (1,11)-polarized abelian surfaces is unirational

We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational but not rational, and there are no Gamma_{11}-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A_{9}^{lev}.Comment: 27 pages, TeX with diagrams.tex. Related Macaulay2 code and PostScript file available at http://www.math.columbia.edu/~psorin

On surfaces in P^4 and 3-folds in P^5

This is a survey on the classification of smooth surfaces in P^4 and smooth 3-folds in P^5. We recall the corresponding results arising from adjunction theory and explain how to construct examples via syzygies. We discuss some examples in detail and list all families of smooth non-general type surfaces in P^4 and 3-folds in P^5 known to us.Comment: 24 pages, AMS-TeX 2.

The moduli space of (111)-polarized abelian surfaces is unirational

We prove that the moduli space $\cal A$11lev of (1,11)-polarized Abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface in P4. Therefore, $\cal A$11lev is unirational but not rational, and there are no Γ11-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of $\cal A$9lev