arXiv subjects

Burt Totaro

Publications and source records attributed to Burt Totaro.

48 records · Page 3Linked to original sources

arXiv2007

Moving codimension-one subvarieties over finite fields

We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero, but no positive multiple of the curve moves in a family of disjoint curves. This answers questions by Keel and Mumford. The proof uses an obstruction theory, in the spirit of homotopy theory, which links the infinitely many obstructions to moving higher and higher multiples of a given codimension-one subvariety. On 3-folds over a finite field, we find nef and big line bundles which are not semi-ample. Finally, we reprove some of the known positive results about semi-ampleness over finite fields.

math.AG↗

arXiv2006

Birational geometry of quadrics in characteristic 2

A conic bundle or quadric bundle in characteristic 2 can have generic fiber which is nowhere smooth over the function field of the base variety. In that case, the generic fiber is called a quasilinear quadric. We solve some of the main problems of birational geometry for quasilinear quadrics. First, if there is a rational map from one quasilinear quadric to another of the same dimension, and also a map back, then the two quadrics are birational. Next, we determine exactly which quasilinear quadrics are ruled, that is, birational over the base field to the product of some variety with the projective line. Both statements are conjectured for quadrics in any characteristic. The proofs begin by extending Karpenko and Merkurjev's theorem on the essential dimension of quadrics to arbitrary quadrics (smooth or not) in characteristic 2.

math.AG↗

arXiv2006

The automorphism group of an affine quadric

We determine the automorphism group for a large class of affine quadric hypersurfaces over a field, viewed as affine algebraic varieties. In particular, we find that the group of real polynomial automorphisms of the n-sphere is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. The proof uses Karpenko's theorem that certain projective quadrics over a field are not ruled. That is, they are not birational over the given field to the product of any variety with the projective line. We also formulate a general result on automorphisms of affine varieties. We conclude by conjecturing a converse to Karpenko's theorem, predicting exactly which projective quadrics are ruled.

math.AG↗

arXiv2004

The curvature of a Hessian metric

Following P. M. H. Wilson's paper on sectional curvatures of Kahler moduli, we consider a natural Riemannian metric on a hypersurface f=1 in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R^3 such that the associated metric has constant negative curvature. Dubrovin found another such polynomial on R^3, the Maschke sextic. We ask if these examples are the only ones with constant negative curvature, and we prove some partial results. This question is related both to the WDVV equations from string theory and to the "Clebsch covariant" from 19th-century invariant theory.

math.DG↗

arXiv2003

Topology of singular algebraic varieties

I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past 20 years have seen a series of new invariants, partly inspired by string theory, such as motivic integration and the elliptic genus of a singular variety. These theories are not defined in a topological way, but there are intriguing hints of their topological significance.

math.AG↗

arXiv2002

Cheeger manifolds and the classification of biquotients

A closed manifold is called a biquotient if it is diffeomorphic to K\G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H. Biquotients are a major source of examples of Riemannian manifolds with nonnegative sectional curvature. We prove several classification results for biquotients: (1) We classify all simply connected rational homology spheres which are diffeomorphic to biquotients. For example, the Gromoll-Meyer exotic sphere is the only exotic sphere of any dimension which is a biquotient. (2) We determine exactly which Cheeger manifolds, the connected sums of two rank-one symmetric spaces, are diffeomorphic to biquotients. For example, CP^2 # CP^2 is a biquotient, but CP^4 # HP^2 is not. (3) There are only finitely many diffeomorphism classes of 2-connected biquotients in each dimension.

math.DG↗

arXiv2002

Curvature, diameter, and quotient manifolds

Gromov showed that there is an upper bound on the Betti numbers of all closed Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether such manifolds (if simply connected) fall into only finitely many rational homotopy types. We give a negative answer, in fact in dimension 6, which is the smallest possible. We also give counterexamples to some related questions in dimensions 7 and 9, improving the original counterexamples by Fang and Rong which were in dimensions at least 22.

math.DG↗

arXiv2002

The resolution property for schemes and stacks

We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme.

math.AG↗

arXiv2000

Chern numbers for singular varieties and elliptic homology

A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so. Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: ``complex bordism modulo flops equals elliptic homology.''

math.AG↗

arXiv1998

The Chow ring of a classifying space

We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to a natural quotient of the complex cobordism ring of the classifying space, a topological invariant. We apply this to get torsion information on the Chow groups of varieties defined as quotients by finite groups. This generalizes Atiyah and Hirzebruch's use of such varieties to give counterexamples to the Hodge conjecture with integer coefficients.

math.AG↗

arXiv1996

Torsion algebraic cycles and complex cobordism

We show that the cycle map on a variety X, from algebraic cycles modulo algebraic equivalence to integer cohomology, lifts canonically to a topologically defined quotient of the complex cobordism ring of X. This more refined cycle map gives a topological proof that the Griffiths group is nonzero for some varieties X, without any use of Hodge theory. We also use this more refined cycle map to give examples of torsion algebraic cycles which map to 0 in Deligne cohomology but are not algebraically equivalent to 0, thus answering some questions by Colliot-Thelene and Schoen.

alg-geom↗

arXiv1993

Initial ideals, Veronese subrings, and rates of algebras

We show that high Veronese subrings of any commutative graded ring have a Grobner basis with all relations of degree 2. (The d-th Veronese subring of a ring A_0 + A_1 + A_2 + ... is the ring A_0 + A_d + A_{2d} + ...; ``high'' means we take d sufficiently large, say at least half the regularity of the ideal defining the original ring.) This gives another proof of Backelin's theorem that such Veronese subrings are Koszul algebras (= wonderful rings), i.e., that the minimal resolution of the residue field of such a ring is linear.

alg-geom↗