Finite Groups of Essential Dimension 2

We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0.Comment: 30 pages (To appear in Commentarii Mathematici Helvetici

Regular pairs of quadratic forms on odd-dimensional spaces in characteristic 2

We describe a normal form for a smooth intersection of two quadrics in even-dimensional projective space over an arbitrary field of characteristic 2. We use this to obtain a description of the automorphism group of such a variety. As an application, we show that every quartic del Pezzo surface over a perfect field of characteristic 2 has a canonical rational point and, thus, is unirational.Comment: 35 pages; v2: introduction expanded with some new theorem

Twisted forms of toric varieties

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M.~Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebrasComment: 41 pages; numerous revisions: introduction more accessible, results now weaker for singular varieties in positive characteristi

Birational self-maps of threefolds of (un)-bounded genus or gonality

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.Comment: 23 pages, some appendix adde