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𝒟 -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

We study the geometry of 𝒟 -bundles—locally projective 𝒟 -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of 𝒟 -bundles; in particular, we prove that the local structure of 𝒟 -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...

A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least 2 . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

A bound for the Milnor number of plane curve singularities

Arkadiusz Płoski (2014)

Open Mathematics

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen (1995)

Annales Polonici Mathematici

Let F = X + H be a cubic homogeneous polynomial automorphism from ℂ n to ℂ n . Let p be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that d e g F - 1 ≤ 3 p - 1 . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

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