arXiv subjects

Herbert Kurke

Publications and source records attributed to Herbert Kurke.

4 recordsLinked to original sources

arXiv2018

Commuting differential operators and higher-dimensional algebraic varieties

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of partial differential operators, and we investigate the properties of these geometric data. This construction is similar to the construction of a formal module of Baker-Akhieser functions. On the other hand, there is a recent generalization of Sato's theory which belongs to the third author of this paper. We compare both approaches to the commutative rings of partial differential operators in two variables.

math.AG↗

arXiv2015

Geometric properties of commutative subalgebras of partial differential operators

We investigate further alebro-geometric properties of commutative rings of partial differential operators continuing our research started in previous articles. In particular, we start to explore the most evident examples and also certain known examples of algebraically integrable quantum completely integrable systems from the point of view of a recent generalization of Sato's theory which belongs to the second author. We give a complete characterisation of the spectral data for a class of "trivial" rings and strengthen geometric properties known earlier for a class of known examples. We also define a kind of a restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to analogous moduli space on a divisor (both the surface and divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding rings of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any commutative ring of PDOs of rank one. At last, we prove that any commutative ring of PDOs, whose normalisation is isomorphic to the ring of polynomials $k[u,t]$, is a Darboux transformation of a ring of operators with constant coefficients.

math.AG↗

arXiv2011

Formal groups arising from formal punctured ribbons

We investigate Picard functor of a formal punctured ribbon. We prove that under some conditions this functor is representable by a formal group scheme. Formal punctured ribbons were introduced in arXiv:0708.0985.

math.AG↗

arXiv2011

Formal punctured ribbons and two-dimensional local fields

We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-to-one correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local field.

math.AG↗