Publications and Preprints

Abstract:

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories. Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.

Abstract:

A line bundle is immaculate if its cohomology vanishes in every dimension. We give a criterion for when a smooth toric Deligne-Mumford stack has infinitely many immaculate line bundles. This answers positively a question of Borisov and Wang.

Abstract:

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: \(\mathfrak{A}_5\), \(\operatorname{PSL}_2(\mathbf{F}_7)\), \(\mathfrak{A}_6\), \(\operatorname{SL}_2(\mathbf{F}_8)\), \(\mathfrak{A}_7\), \(\operatorname{PSp}_4(\mathbf{F}_3)\), \(\operatorname{SL}_2(\mathbf{F}_{7})\), \(2.\mathfrak{A}_5\), \(2.\mathfrak{A}_6\), \(3.\mathfrak{A}_6\) or \(6.\mathfrak{A}_6\). All of these groups with a possible exception of \(2.\mathfrak{A}_6\) and \(6.\mathfrak{A}_6\) indeed act on some rationally connected threefolds.

Abstract:

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.

Abstract:

We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric.

Abstract:

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a slight generalization of this conjecture, where the endomorphism algebras of the exceptional objects are allowed to be separable field extensions of the base field. We show this generalization is false by exhibiting a geometrically rational, smooth, projective threefold over the the field of rational numbers that possesses a full étale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full étale-exceptional collection by a torsor that is invisible to Brauer invariants. Along the way, we develop some tools for linearizing objects, including a group that controls linearizations.

Abstract:

We combine the Bondal-Uehara method for producing exceptional collections on toric varieties with a result of the first author and Favero to expand the set of varieties satisfying Orlov’s Conjecture on derived dimension.

Abstract:

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.

Abstract:

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskiĭ and Klyachko, and toric varieties associated to Weyl fans of type \(A\). Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

Abstract:

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.

Abstract:

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 algebras.

Abstract:

A variety \(X\) with an action of a finite group \(G\) is said to be \(G\)-unirational if there is a \(G\)-equivariant dominant rational map \(V \dasharrow X\) where \(V\) is a faithful linear representation of \(G\). This generalizes the usual notion of unirationality. We determine when \(X\) is \(G\)-unirational for any complex del Pezzo surface \(X\) of degree at least \(3\).

Abstract:

We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces \(X\) with an action of a finite group \(G\) such that \(X\) is equivariantly birational to a surface which has a \(G\)-fixed point.

Abstract:

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety \(X\). Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of \(X\) (existence of rational points, existence of a dense set of rational points, etc.) We give applications of this equivalence in both directions, to study versality of group actions and rational points on algebraic varieties. We obtain similar results on \(p\)-versality for a prime integer \(p\). An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

Abstract:

We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime \(p\) and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man’s essential dimension" of an arbitrary finite group, an intermediate notion between the absolute essential dimension and the essential dimension at a prime \(p\).

Abstract:

We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0.

Abstract:

We show that Y. Prokhorov’s "Simple Finite Subgroups of the Cremona Group of Rank 3" implies that, over any field of characteristic \(0\), the essential dimensions of the alternating group, \(A_7\), and the symmetric group, \(S_7\), are \(4\).

Abstract:

Let \(C_p\) denote the cyclic group of order \(p\), where \(p \ge 3\) is prime. We denote by \(V_n\) the indecomposable \(n\) dimensional representation of \(C_p\) over a field \(\mathbb{F}\) of characteristic \(p\). We compute a set of generators, in fact a SAGBI basis, for the ring of invariants \(\mathbb{F}[\mathbb{V}_2 \oplus \mathbb{V}_2 \oplus \mathbb{V}_3]^{C_p}\).

Abstract:

Let \(k\) be a field, let \(L_n = k[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]\) be the Laurent polynomial ring in \(n\) variables and let \(G\) be a finite group of \(k\)-algebra automorphisms of \(L_n\). We give a necessary and sufficient condition for the ring of invariants \(L_n^G\) to have a SAGBI basis. We show that if this condition is satisfied, then \(L_n^G\) has a SAGBI basis relative to any choice of coordinates in \(L_n\) and any term order.