Lines, conics, and all that

34 pp.This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc

Quantum intersection rings

We examine a few problems of enumerative geometry and present their solutions in the framework of deformed (quantum) cohomology rings.Comment: 73 p, uuencoded, uses harvmac in b mode, 6 figures include

Mirror Symmetry and the Moduli Space for Generic Hypersurfaces in Toric Varieties

The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with $c=9$ whose potential is a sum of $A$-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at $c=9$. We use mirror symmetry to derive the dependence of the models on the complexified K\"ahler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (``twisted'') deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.Comment: 54 pages, use harvmac and epsf, one uuencoded figur

The Genus 0 Gromov-Witten Invariants of Projective Complete Intersections

We describe the structure of mirror formulas for genus 0 Gromov-Witten invariants of projective complete intersections with any number of marked points and provide an explicit algorithm for obtaining the relevant structure coefficients. The structural description alone suffices for some qualitative applications, such as vanishing results and the bounds on the growth of these invariants predicted by R. Pandharipande.Comment: two conjectures added; typos corrected 61 pages, 3 figures, 4 table

Classical Algebraic Geometry

[no abstract available