6,721 research outputs found
Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras
For any , we prove that there exist equivalences between these
apparently unrelated objects: irreducible -dimensional non degenerate
projective varieties different from rational normal
scrolls and 3-covered by twisted cubic curves, up to projective equivalence;
quadro-quadric Cremona transformations of , up to linear
equivalence; -dimensional complex Jordan algebras of rank three, up to
isotopy.
We also provide some applications to the classification of particular classes
of varieties in the class defined above and of quadro-quadric Cremona
transformations, proving also a structure theorem for these birational maps and
for varieties 3-covered by twisted cubics by reinterpreting for these objects
the solvability of the radical of a Jordan algebra.Comment: 30 pages, 1 figure. Corrected typo
The derived category of a non generic cubic fourfold containing a plane
We describe an Azumaya algebra on the resolution of singularities of the
double cover of a plane ramified along a nodal sextic associated to a non
generic cubic fourfold containing a plane. We show that the derived category of
such a resolution, twisted by the Azumaya algebra, is equivalent to the
Kuznetsov component in the semiorthogonal decomposition of the derived category
of the cubic fourfold.Comment: 14 pages, many edits and correction
Moduli Spaces and Formal Operads
Let overline{M}_{g,n} be the moduli space of stable algebraic curves of genus
g with n marked points. With the operations which relate the different moduli
spaces identifying marked points, the family (overline{M}_{g,n})_{g,n} is a
modular operad of projective smooth Deligne-Mumford stacks, overline{M}. In
this paper we prove that the modular operad of singular chains
C_*(overline{M};Q) is formal; so it is weakly equivalent to the modular operad
of its homology H_*(overline{M};Q). As a consequence, the "up to homotopy"
algebras of these two operads are the same. To obtain this result we prove a
formality theorem for operads analogous to Deligne-Griffiths-Morgan-Sullivan
formality theorem, the existence of minimal models of modular operads, and a
characterization of formality for operads which shows that formality is
independent of the ground field.Comment: 36 pages (v3: some typographical corrections
Parametrizing quartic algebras over an arbitrary base
We parametrize quartic commutative algebras over any base ring or scheme
(equivalently finite, flat degree four -schemes), with their cubic
resolvents, by pairs of ternary quadratic forms over the base. This generalizes
Bhargava's parametrization of quartic rings with their cubic resolvent rings
over by pairs of integral ternary quadratic forms, as well as
Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank
2 families of ternary quadratic forms. We give a geometric construction of a
quartic algebra from any pair of ternary quadratic forms, and prove this
construction commutes with base change and also agrees with Bhargava's explicit
construction over .Comment: submitte
On representations of Clifford algebras of ternary cubic forms
In this article, we provide an overview of a one-to-one correspondence
between representations of the generalized Clifford algebra of a ternary
cubic form and certain vector bundles (called Ulrich bundles) on a cubic
surface . We study general properties of Ulrich bundles, and using a recent
classification of Casanellas and Hartshorne, deduce the existence of
irreducible representations of of every possible dimension.Comment: 9 pages, to appear in proceedings for the conference "New Trends in
Noncommutative Algebra: A Conference in Honor of Ken Goodearl's 65th
Birthday
Rings of small rank over a Dedekind domain and their ideals
In 2001, M. Bhargava stunned the mathematical world by extending Gauss's
200-year-old group law on integral binary quadratic forms, now familiar as the
ideal class group of a quadratic ring, to yield group laws on a vast assortment
of analogous objects. His method yields parametrizations of rings of degree up
to 5 over the integers, as well as aspects of their ideal structure, and can be
employed to yield statistical information about such rings and the associated
number fields.
In this paper, we extend a selection of Bhargava's most striking
parametrizations to cases where the base ring is not Z but an arbitrary
Dedekind domain R. We find that, once the ideal classes of R are properly
included, we readily get bijections parametrizing quadratic, cubic, and quartic
rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss
composition for which Bhargava is famous. We expect that our results will shed
light on the analytic distribution of extensions of degree up to 4 of a fixed
number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite
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