Special instanton bundles on ℙ2N+1, their geometry and their moduli

Abstract

O. IntroductionThis paper originated from two quite different generalizations of mathematical instanton bundles on P3(C) and their geometry. Since F2, + 1(C) can be considered as a twister space over the quaternionic manifold IP,(~I), a paper of Salamon, I-S], motivated the construction of holomorphic rank-2n bundles on P2,+1(~) by Okonek and the first author in [OS]. These bundles carry an additional holomorphic symplectic form and are generalizations of the special instanton bundles over P3(~), which are also called special 't Hooft bundles in [HN-J. On the other hand B6hmer and the second author, [BT], found that the latter can be interpreted geometrically by certain Poncel6t curves of higher degree related to certain conics. Such Poncel6t curves had been considered by Darboux in IDa] and I-Ba]. the description of the Poncel6t curves led in IT] to a generalization of Darboux's Poncel6t theorem to the case of hypersurfaces, which are Poncel& related to rational normal curves. In this paper the two kinds of objects are linked together by showing that the special instanton bundles of I'OS] are characterized by Poncel6t hypersurfaees, which are related to linear systems on rational normal curves in a Grassmannian.The first part of this paper on the geometry of special instanton bundles is based on the preprint [ST] and gives a systematic account of these bundles. The second part deals with moduli of special instanton bundles.The main result of the first part is Theorem 5.6. The divisor of jumping lines and the bundle itself are determined by a hypersufface Y, contained in a characteristic 2n-dimensional linear subspace of FA ar 3Gr2C2,+2, and Poncel6t related to the rational normal curve C, which is cut' out of the Grassmannian by the 2n-plane and which describes the set of unstable hyperplanes of the bundle.This result not only generalizes the by now classical description of M(0, 2) by Hartshorne, [H1], in all of its aspects. In the course of proving, all special properties of the special instanton bundles have found their natural interpretation in a geometry, which is closely related to linear systems on rational normal curves and a series of operators between SL(2)-representations related to them.

Homogeneous instanton bundles on P3 for the action of SL(Faenzi1 2007Journal of Geometry and Physics2200View full textAdd to dashboardCite

“…Special instantons were first studied in [10], and have been extensively investigated ever since, see e.g. [14,13], so we will say no more about them here.…”

Complex ADHM equations and sheaves on P3

“…This means that rank 2 locally free instanton sheaves are mathematical instanton bundles on P 3 , an object first introduced in [17,18] and later studied by many authors (see for instance [1,[3][4][5]14,19]). However, almost all of the literature on instanton bundles has concentrated on locally-free, rank 2 bundles.…”

“…Many authors have studied unframed rank 2n instanton bundles over P 2n+1 over the past 25 years, see for instance [1,3,14,18,19]. Establishing irreducibility and smoothness of their moduli space and computing its dimension has proved to be a very hard problem; the introduction of [4] has a nice survey on this topic, as well as all relevant references.…”

Liaison, Schottky Problem and Invariant Theory

“…A similar treatment of the osculating spaces to R N needed to studying the divisors on the curve is done in [SpT1] and [SpT2] as an approach to certain vector bundles.…”