Curves having one place at infinity and linear systems on rational surfaces

Denoting by ${\mathcal L}_d(m_0,m_1,...,m_r)$ the linear system of plane curves passing through $r+1$ generic points $p_0,p_1,...,p_r$ of the projective plane with multiplicity $m_i$ (or larger) at each $p_i$, we prove the Harbourne-Hirschowitz Conjecture for linear systems ${\mathcal L}_d(m_0,m_1,...,m_r)$ determined by a wide family of systems of multiplicities $\bold{m}=(m_i)_{i=0}^r$ and arbitrary degree $d$. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system $\bold{m}$ and we give its exact value when $\bold{m}$ is in the above family. To do that, we prove an $H^1$-vanishing theorem for line bundles on surfaces associated with some pencils ``at infinity''.Comment: This is a revised version of a preprint of 200

Characterizations of the Suzuki tower near polygons

In recent work, we constructed a new near octagon $\mathcal{G}$ from certain involutions of the finite simple group $G_2(4)$ and showed a correspondence between the Suzuki tower of finite simple groups, $L_3(2) < U_3(3) < J_2 < G_2(4) < Suz$, and the tower of near polygons, $\mathrm{H}(2,1) \subset \mathrm{H}(2)^D \subset \mathsf{HJ} \subset \mathcal{G}$. Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon $\mathsf{HJ}$ is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters $(2, 4; 0, 3)$, but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, $\mathrm{H}(2)^D$. We also give a complete classification of near hexagons of order $(2, 2)$ and use it to prove the uniqueness result for $\mathrm{H}(2)^D$.Comment: 20 pages; some revisions based on referee reports; added more references; added remarks 1.4 and 1.5; corrected typos; improved the overall expositio

Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory

The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry and references adde

Tropical curves, graph complexes, and top weight cohomology of M_g

We study the topology of a space parametrizing stable tropical curves of genus g with volume 1, showing that its reduced rational homology is canonically identified with both the top weight cohomology of M_g and also with the genus g part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least 7. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of another theorem of Willwacher, that homology of the graph complex vanishes in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J. Amer. Math. So

A note on the spectral mapping theorem of quantum walk models

We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of $U$ without the spectral decomposition of $T$. Arguments in this direction reveal the eigenspaces of $U$ characterized by the generalized kernels of linear operators given by $T$.Comment: 17 page

A new near octagon and the Suzuki tower

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the group $\mathrm{G}_2(4){:}2$ and which contains $416$ copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the $\mathrm{G}_2(4)$-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is $(2,4)$.Comment: 24 pages, revised version with added remarks and reference

Non-normal affine monoids

We give a geometric description of the set of holes in a non-normal affine monoid $Q$. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of $k[Q]$. From this, we see how various properties of $k[Q]$ like local normality and Serre's conditions $(R_1)$ and $(S_2)$ are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra $k[Q]$ is obtained and some cases where equality holds are identified. We apply this results to seminormal affine monoids.Comment: 18 pages, 3 figures. Simplified proof of the main result, shortened. An even shorter version appeared with the title "Non-normal affine monoid algebra" in manuscripta mathematic