36 research outputs found
Likelihood Geometry
We study the critical points of monomial functions over an algebraic subset
of the probability simplex. The number of critical points on the Zariski
closure is a topological invariant of that embedded projective variety, known
as its maximum likelihood degree. We present an introduction to this theory and
its statistical motivations. Many favorite objects from combinatorial algebraic
geometry are featured: toric varieties, A-discriminants, hyperplane
arrangements, Grassmannians, and determinantal varieties. Several new results
are included, especially on the likelihood correspondence and its bidegree.
These notes were written for the second author's lectures at the CIME-CIRM
summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition
Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions
We present a formula for a classical -matrix of an integrable system
obtained by Hamiltonian reduction of some free field theories using pure gauge
symmetries. The framework of the reduction is restricted only by the assumption
that the respective gauge transformations are Lie group ones. Our formula is in
terms of Dirac brackets, and some new observations on these brackets are made.
We apply our method to derive a classical -matrix for the elliptic
Calogero-Moser system with spin starting from the Higgs bundle over an elliptic
curve with marked points. In the paper we also derive a classical
Feigin-Odesskii algebra by a Poisson reduction of some modification of the
Higgs bundle over an elliptic curve. This allows us to include integrable
lattice models in a Hitchin type construction.Comment: 27 pages LaTe
The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses
This survey may be seen as an introduction to the use of toric and tropical
geometry in the analysis of plane curve singularities, which are germs
of complex analytic curves contained in a smooth complex analytic surface .
The embedded topological type of such a pair is usually defined to be
that of the oriented link obtained by intersecting with a sufficiently
small oriented Euclidean sphere centered at the point , defined once a
system of local coordinates was chosen on the germ . If one
works more generally over an arbitrary algebraically closed field of
characteristic zero, one speaks instead of the combinatorial type of .
One may define it by looking either at the Newton-Puiseux series associated to
relative to a generic local coordinate system , or at the set of
infinitely near points which have to be blown up in order to get the minimal
embedded resolution of the germ or, thirdly, at the preimage of this
germ by the resolution. Each point of view leads to a different encoding of the
combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques
diagram, or a weighted dual graph. The three trees contain the same
information, which in the complex setting is equivalent to the knowledge of the
embedded topological type. There are known algorithms for transforming one tree
into another. In this paper we explain how a special type of two-dimensional
simplicial complex called a lotus allows to think geometrically about the
relations between the three types of trees. Namely, all of them embed in a
natural lotus, their numerical decorations appearing as invariants of it. This
lotus is constructed from the finite set of Newton polygons created during any
process of resolution of by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is
new. The historical information, contained before in subsection 6.2, is
distributed now throughout the paper in the subsections called "Historical
comments''. More details are also added at various places of the paper. To
appear in the Handbook of Geometry and Topology of Singularities I, Springer,
202
Compactifications of Moduli of Points and Lines in the Projective Plane
Projective duality identifies the moduli spaces B-n and X(3, n) parametrizing linearly general configurations of n points in P-2 and n lines in the dual P-2, respectively. The space X(3, n) admits Kapranov's Chow quotient comp actification (X) over bar (3, n), studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of P-2 with n "broken lines". Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of P-2 with n smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003
Recommended from our members
Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces
173-22