Michele Bolognesi

Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

Abstract Let X → Y be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools—a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting—to study semiorthogonal decompositions of the bounded derived category D b ( X ) . Together with results in the theory of quadratic forms, we apply these tools in the case where r = 2 and X → Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus one, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if Y = P 1 over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.

Stacks of trigonal curves

In this paper we study the stack Tg of smooth triple covers of a conic; when g � 5 this stack is embedded Mg as the locus of trigonal curves. We show that Tg is a quotient (Ug/ g), where g is a certain algebraic group and Ug is an open subscheme of a g-equivariant vector bundle over an open subscheme of a representation of g. Using this, we compute the integral Picard group of Tg when g > 1. The main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym 3 E det E _ , and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack. In moduli theory, stacks have often been constructed as quotients by group actions. If an algebraic stack F is a quotient stack (X=G), where G is an algebraic group acting on an algebraic variety X, the geometry of F is the geometry of the action of G on X, and one can apply to F the powerful techniques that have been developed for studying invariants of group actions in algebraic topology and algebraic geometry. Even just knowing that such a presentation exists, even without an explicit description of the action, can be useful: but it is even better when the variety X and the group G are fairly simple, so that this description may be used directly to study F. This does not seem possible in many cases: for example, the stack Mg of smooth curves of genus g is of the form (X=G), but when g is large the space X is complicated, and to our knowledge no general result about Mg has been proved by exploiting this presentation. (Of course, in Teichmuller theory one represents Mg as a quotient of an action of the Teichmuller group, which is an infinite discrete group, acting on a ball in C 3g−3 , but this description is topological, and it is hard to use it directly to prove algebraic geometric results about Mg.) It has long been known that in characteristic different from 2 and 3, the stack M1;1 of elliptic curves is a quotient ((X=Gm)), where X is the complement of the hypersurface 4x 3 + 27y 2 = 0 in A 2 , and Gm acts with weights 4 and 6. This gives an easy proof of the fact, due to Mumford ((Mum65)), that the Picard group of M1;1 is cyclic of order 12. In (Vis98), the second author gives a presentation of M2 as a quotient (X=GL2), where X is the scheme of smooth binary forms of degree 6 in two variables (the action of GL2 on X is a twist of the customary one). As an application he computes the Chow ring of M2. This was generalized in (AV04) to the stack Hg of hyperelliptic curves of genus g, which has a presentation as a quotient (Xg=GL2) (if g is even), or (Xg=(GmPGL2)) (if g is odd), where X is the space of smooth binary forms of degree 2g + 2 in two variables; this allows

Derived categories and rationality of conic bundles.

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

CUBIC FOURFOLDS CONTAINING A PLANE AND A QUINTIC DEL PEZZO SURFACE

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Cliord algebra over the K3 surface S of degree 2 arising from X. Specically, we show that in the moduli space of cubic fourfolds, the intersection of divisorsC8\C14 has ve irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaan and has nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov’s derived categorical conjecture on the rationality of cubic fourfolds.

Categorical representability and intermediate Jacobians of Fano threefolds

We define, basing upon semiorthogonal decompositions of

On Weddle surfaces and their moduli

\Db(X)

SOME LOCI OF RATIONAL CUBIC FOURFOLDS

, categorical representability of a projective variety

Factorization of point configurations, cyclic covers, and conformal blocks

X

Forgetful linear systems on the projective space and rational normal curves over ℳGIT0,2n

and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of

MODULI OF ABELIAN SURFACES, SYMMETRIC THETA STRUCTURES AND THETA CHARACTERISTICS

X