31,967 research outputs found
Dynamical Cusp Regeneration
After being destroyed by a binary supermassive black hole, a stellar density
cusp can regrow at the center of a galaxy via energy exchange between stars
moving in the gravitational field of the single, coalesced hole. We illustrate
this process via high-accuracy N-body simulations. Regeneration requires
roughly one relaxation time and the new cusp extends to a distance of roughly
one-fifth the black hole's influence radius, with density rho ~ r^{-7/4}; the
mass in the cusp is of order 10% the mass of the black hole. Growth of the cusp
is preceded by a stage in which the stellar velocity dispersion evolves toward
isotropy and away from the tangentially-anisotropic state induced by the
binary. We show that density profiles similar to those observed at the center
of the Milky Way and M32 can regenerate themselves in several Gyr following
infall of a second black hole; the presence of density cusps at the centers of
these galaxies can therefore not be used to infer that no merger has occurred.
We argue that Bahcall-Wolf cusps are ubiquitous in stellar spheroids fainter
than M_V ~ -18.5 that contain supermassive black holes, but the cusps have not
been detected outside of the Local Group since their angular sizes are less
than 0.1". We show that the presence of a cusp implies a lower limit of
\~10^{-4} per year on the rate of stellar tidal disruptions, and discuss the
consequences of the cusps for gravitational lensing and the distribution of
dark matter on sub-parsec scales.Comment: Accepted for publication in The Astrophysical Journa
A modular description of
As we explain, when a positive integer is not squarefree, even over
the moduli stack that parametrizes generalized elliptic curves
equipped with an ample cyclic subgroup of order does not agree at the cusps
with the -level modular stack defined by
Deligne and Rapoport via normalization. Following a suggestion of Deligne, we
present a refined moduli stack of ample cyclic subgroups of order that does
recover over for all . The resulting modular
description enables us to extend the regularity theorem of Katz and Mazur:
is also regular at the cusps. We also prove such regularity
for and several other modular stacks, some of which have
been treated by Conrad by a different method. For the proofs we introduce a
tower of compactifications of the stack that
parametrizes elliptic curves---the ability to vary in the tower permits
robust reductions of the analysis of Drinfeld level structures on generalized
elliptic curves to elliptic curve cases via congruences.Comment: 67 pages; final version, to appear in Algebra and Number Theor
On quartics with three-divisible sets of cusps
We study the geometry and codes of quartic surfaces with many cusps. We apply
Gr\"obner bases to find examples of various configurations of cusps on
quartics.Comment: 15 page
Cusps and Codes
We study a construction, which produces surfaces with cusps.
For example we obtain surfaces of degree six with 18, 24 or 27 three-divisible
cusps. For sextic surfaces in a particular family of up to 30 cusps the codes
of these sets of cusps are determined explicitly.Comment: 13 page
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