Jet Bundles on Gorenstein Curves and Applications

Journal of Singularities
volume 21 (2020), 50-83

Received: 22 April 2018. Received in revised form: 13 September 2018.

DOI: 10.5427/jsing.2020.21d

Add a reference to this article to your citeulike library.

Abstract:

The purpose of this expository paper is to present a catalogue of locally free replacements of the sheaves of principal parts for (families of) Gorenstein curves. In the smooth category, locally free sheaves of principal parts are better known as jet bundles, understood as those locally free sheaves whose transition functions reflect the transformation rules of the partial derivatives of a local section under a change of local coordinates. Being a natural globalisation of the fundamental notion of Taylor expansion of a function in a neighborhood of a point, jet bundles are ubiquitous in Mathematics. They proved powerful tools for the study of deformation theories within a wide variety of mathematical situations and have a number of purely algebraic incarnations: besides the aforementioned principal parts of a quasi-coherent sheaf we should mention, for instance, the theory of arc spaces on algebraic varieties, introduced by Nash to deal with resolutions of singular loci of singular varieties.

The issue we want to cope with in this survey is that sheaves of principal parts of vector bundles defined on a singular variety X are not locally free. Roughly speaking, the reason is that the analytic construction carried out in the smooth category, based on gluing local expressions of sections together with their partial derivatives, up to a given order, is no longer available. Indeed, around singular points there are no local parameters with respect to which one can take derivatives. This is yet another way of saying that the sheaf Omega^1_X of sections of the cotangent bundle is not locally free at the singular points.

If C is a projective reduced singular curve, it is desirable, in many interesting situations, to dispose of a notion of global derivative of a regular section. If the singularities of C are mild, that is, if they are Gorenstein, locally free substitutes of the classical principal parts can be constructed by exploiting a natural derivation O_C -> omega_C, taking values in the dualising sheaf, which by the Gorenstein condition is an invertible sheaf. This allows one to mimic the usual procedure adopted in the smooth category. Related constructions have recently been reconsidered by A. Patel and A. Swaminathan, under the name of sheaves of invincible parts, motivated by the classical problem of counting hyperflexes in one-parameter families of plane curves. Besides, locally free jets on Gorenstein curves have been investigated by a number of authors, starting about twenty years ago. The reader can consult other references for several applications.

Author(s) information: