113 research outputs found
Torsion points on the cohomology jump loci of compact K\"ahler manifolds
We prove that each irreducible component of the cohomology jump loci of rank
one local systems over a compact K\"ahler manifold contains at least one
torsion point. This generalizes a theorem of Simpson for smooth complex
projective varieties. An immediate consequence is the conjecture of Beauville
and Catanese for compact K\"ahler manifolds. We also provide an example of a
compact K\"ahler manifold, whose cohomology jump loci can not be realized by
any smooth complex projective variety.Comment: Final version. To appear in Math. Res. Let
Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the
following theorem of de Bruijn and Erd\H{o}s: Every set of points in a
projective plane determines at least lines, unless all the points are
contained in a line. Motzkin and others extended the result to higher
dimensions, who showed that every set of points in a projective space
determines at least hyperplanes, unless all the points are contained in a
hyperplane. Let be a spanning subset of a -dimensional vector space. We
show that, in the partially ordered set of subspaces spanned by subsets of ,
there are at least as many -dimensional subspaces as there are
-dimensional subspaces, for every at most . This confirms the
"top-heavy" conjecture of Dowling and Wilson for all matroids realizable over
some field. The proof relies on the decomposition theorem package for
-adic intersection complexes.Comment: 18 pages, major revisio
Bounding the maximum likelihood degree
Maximum likelihood estimation is a fundamental computational problem in
statistics. In this note, we give a bound for the maximum likelihood degree of
algebraic statistical models for discrete data. As usual, such models are
identified with special very affine varieties. Using earlier work of Franecki
and Kapranov, we prove that the maximum likelihood degree is always less or
equal to the signed intersection-cohomology Euler characteristic. We construct
counterexamples to a bound in terms of the usual Euler characteristic
conjectured by Huh and Sturmfels.Comment: v2: final version, to appear in Math. Res. Let
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