41 research outputs found
Reduced word enumeration, complexity, and randomization
A reduced word of a permutation is a minimal length expression of as
a product of simple transpositions. We examine the computational complexity,
formulas and (randomized) algorithms for their enumeration. In particular, we
prove that the Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, is
typically exponentially large. This implies a result of B. Pawlowski, that it
has exponentially growing expectation. Our result is established by a formal
run-time analysis of A. Lascoux-M.-P. Sch\"utzenberger's transition algorithm.
The more general problem of Hecke word enumeration, and its closely related
question of counting set-valued standard Young tableaux, is also investigated.
The latter enumeration problem is further motivated by work on Brill-Noether
varieties due to M. Chan-N. Pflueger and D. Anderson-L. Chen-N. Tarasca.Comment: 23 pages. v2: added reference to University of Washington PhD thesis
of B. Pawlowski, which proves a stronger version of Theorem 1.
Linear series on metrized complexes of algebraic curves
A metrized complex of algebraic curves is a finite metric graph together with
a collection of marked complete nonsingular algebraic curves, one for each
vertex, the marked points being in bijection with incident edges. We establish
a Riemann-Roch theorem for metrized complexes of curves which generalizes both
the classical Riemann-Roch theorem and its graph-theoretic and tropical
analogues due to Baker-Norine, Gathmann-Kerber, and Mikhalkin-Zharkov. We also
establish generalizations of the second author's specialization lemma and its
weighted graph analogue due to Caporaso and the first author, showing that the
rank of a divisor cannot go down under specialization from curves to metrized
complexes. As an application of these considerations, we formulate a
generalization of the Eisenbud-Harris theory of limit linear series to
semistable curves which are not necessarily of compact type.Comment: Major revision taking into account comments of the referees, e.g.,
proofs are shortened and clarified, notations improved, changes in
organization of the sections, etc. 45 page
Geometry of intersections of some secant varieties to algebraic curves
For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulae counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.Peer Reviewe
The Kodaira dimensions of and
We prove that the moduli spaces of curves of genus 22 and 23 are of general
type. To do this, we calculate certain virtual divisor classes of small slope
associated to linear series of rank 6 with quadric relations. We then develop
new tropical methods for studying linear series and independence of quadrics
and show that these virtual classes are represented by effective divisors.Comment: 94 pages, 27 figures; incorporates and supersedes arXiv:1804.01898
and arXiv:1808.0128
Scattering amplitudes of stable curves
Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of
stable rational curves, introduced by Castravet and Tevelev, appear as
numerators of scattering amplitude forms for n massless particles in N=4
Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov
and Trnka. Rather than being a coincidence, this is just the tip of the iceberg
of an exciting relation between algebraic geometry and high energy physics. We
interpret leading singularities of scattering amplitude forms of massless
particles as probabilistic Brill-Noether theory: the study of statistics of
images of n marked points under a random meromorphic function uniformly
distributed with respect to the translation-invariant volume form of the
Jacobian. We focus on the maximum helicity violating regime, which leads to a
beautiful physics-inspired geometry for various classes of algebraic curves:
smooth, stable, hyperelliptic, real algebraic, etc.Comment: 54 pages. A small section added on the non-MHV case and relation to
Grassmannian amplitude