Related papers: On the rationality problem for quadric bundles

We construct a family of non-collapsed, non-K\"ahler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over K\"ahler-Einstein manifolds of positive scalar…

By the technique of 3-fold Mori theory, we prove that the moduli space whose general point parameterizes a couple of a smooth curve of genus 4 and a halfcanonical divisor with vanishing global section is rational.

This is a continuation of "Rational families of vector bundles on curves, I". Let C be a smooth projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1.…

We show that a very general (3,3) complete intersection in $\mathbb{P}^7$ over an algebraically closed uncountable field of characteristic different from 2 admits no decomposition of the diagonal, in particular it is not retract rational.…

The moduli space M(r,d) of stable, rank r, degree d vector bundles on a smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A, where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The birational…

Let $X$ be a smooth projective manifold with $\dim_\mathbb{C} X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti-Grauert theorem. As an application, we show…

We study the essential dimension and essential p-dimension of the moduli stack of vector bundles over a smooth orbifold curve containing a rational point. We improve the known bounds on this essential dimension and obtain an equality modulo…

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least…

Consider the blow up $\pi: \widetilde{X} \to X$ of a rational surface $X$ at a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and…

Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincar\'e…

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We…

We show that rationality does not specialize in flat projective families of complex fourfolds with terminal singularities. This answers a question of Totaro, who established the analogous result in all dimensions greater than 4.

We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than (2n-2). In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, and prove their…

This is a revised version of the second half of my paper math.AG/9909021. We prove that there exists a positive integer $\nu_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over complex…

We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of 1-cycles on a cubic hypersurface is universally generated by lines. Applications are mainly…

We prove that for any $r \geq 2$ the moduli space of stable Ulrich bundles of rank $r$ and determinant $\mathcal O_X(r)$ on any smooth Fano threefold $X$ of index two is smooth of dimension $r^2+1$ and that the same holds true for even $r$…

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

We show that a very general hypersurface of degree d at least 4 and dimension at most $(d+1)2^{d-4}$ over a field of characteristic different from 2 does not admit a decomposition of the diagonal; hence, it is neither stably nor retract…

The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.

Let $C$ be a curve of genus at least three defined over a number field, and let $r$ be the rank of the rational points of its Jacobian. Under mild hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the…