8,658 research outputs found
Conservative descent for semi-orthogonal decompositions
Motivated by the local flavor of several well-known semi-orthogonal
decompositions in algebraic geometry, we introduce a technique called
conservative descent, which shows that it is enough to establish these
decompositions locally. The decompositions we have in mind are those for
projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due
to Ishii and Ueda. Our technique simplifies the proofs of these decompositions
and establishes them in greater generality for arbitrary algebraic stacks.Comment: Final versio
Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities
Let X be a surface with an isolated singularity at the origin, given by the
equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In
particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C)
finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson
homology of Y, as a graded vector space with respect to the weight grading. In
the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson
homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth
Hochschild homology of the n-th symmetric power of the algebra of G-invariant
differential operators on C. That is, the Brylinski spectral sequence
degenerates in this case. In the elliptic case, this yields the zeroth
Hochschild homology of symmetric powers of the elliptic algebras with three
generators modulo their center, for the parameter equal to all but countably
many points of the elliptic curve.Comment: 17 page
Noncommutative curves and noncommutative surfaces
In this survey article we describe some geometric results in the theory of
noncommutative rings and, more generally, in the theory of abelian categories.
Roughly speaking and by analogy with the commutative situation, the category
of graded modules modulo torsion over a noncommutative graded ring of
quadratic, respectively cubic growth should be thought of as the noncommutative
analogue of a projective curve, respectively surface. This intuition has lead
to a remarkable number of nontrivial insights and results in noncommutative
algebra. Indeed, the problem of classifying noncommutative curves (and
noncommutative graded rings of quadratic growth) can be regarded as settled.
Despite the fact that no classification of noncommutative surfaces is in sight,
a rich body of nontrivial examples and techniques, including blowing up and
down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis
Keeler) have been incorporated. The formulation of some results has been
improve
Some Global Characteristics of the Galactic Globular Cluster System
The relations between the luminosities , the metallicities ,
the Galactocentric radii , and the central concentration indices of
Galactic globular clusters are discussed. It is found that the most luminous
clusters rarely have collapsed cores. The reason for this might be that the
core collapse time scales for such populous clusters are greater than the age
of the Galaxy. Among those clusters, for which the structure has not been
modified by core collapse, there is a correlation between central concentration
and integrated luminosity, in the sense that the most luminous clusters have
the strongest central concentration. The outermost region of the Galaxy with
kpc was apparently not able to form metal-rich globular
clusters, whereas such clusters (of which Ter 7 is the prototype) were able to
form in some nearby dwarf spheroidal galaxies. It is not yet clear how the
popular hypothesis that globular clusters were initially formed with a single
power law mass spectrum can be reconciled with the observation that both (1)
Galactic globular clusters with kpc, and (2) the globulars associated
with the Sagittarius dwarf, appear to have bi-modal luminosity functions.Comment: 15 pages, 1 figur
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