We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projec...

There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ≥ f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois gro...

We show that every tilting module of projective dimension one over a ring R is associated in a natural way to the universal localization R → RU at a set U of finitely presented modules of projective dimension one. We then investigate tilting modules of the form RU ⊕ RU/R. Furthermore, we discuss the relationship between universal localization and the localization R → QG given by a perfect Gabri...

The first examples of complete projective connections are uncovered: on surfaces, normal projective connections whose geodesics are all closed and embedded are complete. On manifolds of any dimension, normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature are complete.

Let R be a ring and M a right R-module. M is called n-FP-projective if Ext M N = 0 for any right R-module N of FP-injective dimension ≤n, where n is a nonnegative integer or n = . R M is defined as sup n M is n-FP-projective and R M = −1 if Ext M N = 0 for some FP-injective right R-module N. The right -dimension r -dim R of R is defined to be the least nonnegative integer n such that R M ≥ n im...

We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if Torn (Ω, k) = 0 for all n = 1, . . . , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor1 (Ω, k) = ...

We answer the first non-classical case of a question of J. Harris from the 1983 ICM: what is the largest possible dimension of a complete subvariety of Mg ? Working over a base field with characteristic 0 or greater then 3 we prove that there are no projective surfaces in the moduli space of curves of genus 4; thus proving that the largest possible dimension of a projective subvariety in M4 is 1.

A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface. In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dol...

There exists a function f : N → N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension ≥ f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois gro...

We answer the first non-classical case of a question of J. Harris from the 1983 ICM: what is the largest possible dimension of a complete subvariety of Mg ? Working over a base field with characteristic 0, we prove that there are no projective surfaces in the moduli space of curves of genus 4; thus proving that the largest possible dimension of a projective subvariety in M4 is 1.