133 research outputs found
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
Numerical invariants and moduli spaces for line arrangements
Using several numerical invariants, we study a partition of the space of line
arrangements in the complex projective plane, given by the intersection lattice
types. We offer also a new characterization of the free plane curves using the
Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.Comment: v3: A new proof of a result due to Tohaneanu, giving the
classification of line arrangements with a Jacobian syzygy of minimal degree
2 is given in Theorem 4.11. Some other minor change
Character sums associated to finite Coxeter groups
We prove a character sum identity for Coxeter arrangements which is a finite
field analogue of Macdonald's conjecture proved by Opdam.Comment: 20 pages, to appear in Transactions of the A.M.
On properness of K-moduli spaces and optimal degenerations of Fano varieties
We establish an algebraic approach to prove the properness of moduli spaces
of K-polystable Fano varieties and reduce the problem to a conjecture on
destabilizations of K-unstable Fano varieties. Specifically, we prove that if
the stability threshold of every K-unstable Fano variety is computed by a
divisorial valuation, then such K-moduli spaces are proper. The argument relies
on studying certain optimal destabilizing test configurations and constructing
a Theta-stratification on the moduli stack of Fano varieties.Comment: v2: to appear in Selecta Mat
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