Daniel Allcock

The complex hyperbolic geometry of the moduli space of cubic surfaces

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Fermat cubic surface. Related results, not quite as extensive, were announced in alg-geom/9709016.

Braid pictures for Artin groups

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A n , B n = C n and D n and the affine diagrams A n , B n , C n and D n as subgroups of the braid groups of various simple orbifolds. The cases D n , B n , C n and D n are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except A n the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type D n .

The moduli space of cubic threefolds

We describe the moduli space of cubic hypersurfaces in CP 4 in the sense of geometric invariant theory. That is, we characterize the stable and semistable hypersurfaces in terms of their singularities, and determine the equivalence classes of semistable hypersurfaces under the equivalence relation of their orbit-closures meeting.

New complex- and quaternion-hyperbolic reflection groups

We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we find reflection groups of finite index. These provide new finite-covolume reflection groups acting on complex and quaternionic hyperbolic spaces. Specifically, we provide groups acting on CH^n for all n<6 and n=7, and on HH^n for n=1,2,3 and 5. We compare our groups to those discovered by Deligne and Mostow and show that our largest examples are new. For many of these Lorentzian lattices we show that the entire symmetry group is generated by reflections, and obtain a description of the group in terms of the combinatorics of a lower-dimensional positive-definite lattice. The techniques needed for our lower-dimensional examples are elementary, but to construct our best examples we also need certain facts about the Leech lattice. We give a new and geometric proof of the classifications of selfdual Eisenstein lattices of dimension < 7 and of selfdual Hurwitz lattices of dimension < 5.

A complex hyperbolic structure for moduli of cubic surfaces

Abstract We show that the moduli space M of marked cubic surfaces is biholomorphic to ( B 4 − H )/Г, where B 4 is complex hyperbolic four-space, Γ is a specific group generated by complex reflections, and H is the union of reflection hyperplanes for Γ. Thus M has a complex hyperbolic structure, i.e., an (incomplete) metric of constant negative holomorphic sectional curvature.

Infinitely many hyperbolic Coxeter groups through dimension 19

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinct Coxeter groups in H^n with noncompact fundamental domain of volume less than or equal to V grows at least exponentially with respect to V. The same result holds for cocompact groups for n < 7. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.

The period lattice for Enriques surfaces

We simplify the usual statement of the Torelli theorem for complex Enriques surfaces, by means of a lattice-theoretic trick. This allows easy proofs of several known results, which previously required intricate arithmetic arguments. The main new result is that the moduli space has contractible universal cover.

Steinberg groups as amalgams

For any root system and any commutative ring we give a relatively simple presentation of a group related to its Stein- berg group St. This includes the case of innite root systems used in Kac-Moody theory, for which the Steinberg group was dened by Tits and Morita-Rehmann. In many cases our group equals St, giving a presentation with many advantages over the usual presen- tation of St. This equality holds for all spherical root systems, all irreducible ane root systems of rank > 2, and all 3-spherical root systems. When the coecient ring satises a minor condition, the last hypothesis can be relaxed to 2-sphericity. Our presentation is dened in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coecients or signs. It makes manifest the exceptional diagram au- tomorphisms in characteristics 2 and 3, and their generalizations to Kac-Moody groups. And it is a Curtis-Tits style presentation: it is the direct limit of the groups coming from 1- and 2-node sub- diagrams of the Dynkin diagram. Over non-elds this description as a direct limit is new and surprising, even for nite root systems. We use it to show that many Steinberg and Kac-Moody groups over nitely-generated rings are nitely presented.

A BANACH SPACE DETERMINED BY THE WEIL HEIGHT

The absolute logarithmic Weil height is well defined on the quo- tient group Q ◊ /Tor Q ◊ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L 1 (Y,B, ), where Y is a certain totally disconnected, locally compact space, B is the -algebra of Borel subsets of Y , and is a certain measure satisfying an invariance property with respect to the absolute Galois group Aut(Q/Q).

Presentation of affine Kac–Moody groups over rings

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.