Canonical Representations of Orthogonal Groups of Line-Bundle-Valued Ternary Quadratic Bundles over Schemes with Applications -Dedicated to Professor Martin Kneser
by
Venkata Balaji, Thiruvalloor Eesanaipaadi
Mathematisches Institut Georg-August-Universitaet Goettingen

Given a line-bundle-valued ternary quadratic bundle over any scheme, there is a functorial representation of its group of orthogonal similitudes in the Witt-invariant, which by definition is the degree zero part of the associated generalised Clifford algebra bundle. Local computations involving twisted even-exterior algebra bundles of rank 3 vector bundles and global scheme-theoretic methods lead to the canonical determination of this representation. The use of the notion of semiregularity introduced by Martin Kneser allows working over arbitrary schemes, regardless of the characteristics of the residue fields of its points. The applications of this computation are as follows.

Application 1: Degeneration Theory of Ternary Quadratic Forms and Rank 4 Azumaya Bundles:

The association of the isomorphism class of a quadratic bundle to its Witt-invariant induces a natural bijection, from the set of equivalence classes of line-bundle-valued quadratic forms on rank 3 vector bundles isometric up to tensoring by twisted discriminant bundles, to the set of isomorphism classes of schematic specialisations of rank 4 Azumaya bundles over any fixed scheme X. This statement is a limiting version of the following theorem of Max-Albert Knus in cohomology: the set of orbits of Disc(X) in the 1-cohomology of X in the fppf topology with values in O(3) is in bijection with the 1-cohomology with values in PGL(2).

The various orthogonal groups of a quadratic bundle are canonically determined in terms of the automorphisms of its even Clifford algebra. Any automorphism of the latter arises from a similarity, and in fact from an orthogonal transformation if its determinant is a square. The special orthogonal group is thus identified with the subgroup of automorphisms with trivial determinant. If X is integral and the quadratic form is semiregular at some point of X, then every automorphism of the even Clifford algebra has determinant 1 and is thus induced from a self-isometry; the orthogonal group is also seen to be a semidirect product in this case.

A specialised algebra arises from a honest quadratic form iff its determinant has a square root and arises from a bilinear form iff the line subbundle generated by 1 is a direct summand.

For a connected proper scheme of finite type over an algebraically closed field, the hypothesis of self-duality on a unital associative algebra bundle of square rank forces the algebra to be either globally Azumaya or to be nowhere-Azumaya. Hence the self-duality of the underlying bundle of the even Clifford algebra implies that the quadratic bundle is semiregular everywhere if it is semiregular even at a single point.

Application 2: Geometry of the Scheme of Specialisations of Azumaya Algebra Structures on a rank 4 vector bundle and applications to Desingularisation

The multiplication table of every specialised algebra structure on any fixed free rank 4 vector bundle with fixed unit that is part of a global basis is computed explicitly. The key theorem is that the natural parameter space for specialisations of Azumaya algebra structures on a fixed (locally-free) rank 4 vector bundle is smooth and geometrically irreducible over the base scheme. (It is known that the smoothness does not hold for other higher ranks). This generalises theorems of Seshadri and Ramanan.

The smoothness result allows the construction of a generalised Seshadri desingularisation of the moduli space of semistable rank 2 degree zero vector bundles on a curve relative to a locally-universally-japanese (Nagata) base scheme.

It also allows the construction of a generalised Nori desingularisation of the Artin moduli space of invariants of several matrices in rank 2 over a locally-Nagata base scheme, along with the good specialisation property over the integers.

When the base scheme is the spectrum of an algebraically closed field, a canonical stratification of the variety of specialisations is obtained.

Keywords:

Vector Bundle, Desingularisation, Azumaya bundle, bilinear form, Clifford algebra, discriminant bundle, line-bundle-valued form, orthogonal group, quadratic bundle, quaternion bundles, semiregular form, similarity, similitude, ternary form, Witt-invariant.

MSC Subject Classification Numbers:

14A25, 14F05, 14L15, 14M, 14Q, 15A63, 15A66, 15A75, 15A78, 16H05, 16S60, 16W20, 20G05, 20G35

Date received: January 26, 2005