Jumping numbers on algebraic surfaces with rational singularities

Abstract

Abstract. In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

Jumping numbers and ordered tree structures on the dual graph

“…The associated proximity matrix and its inverse are clearly restrictions of those of (10). So k ′ ν = k ν for every ν = N. Because N is adjacent only to β and E 2 N = −1, the proximity equation for the ideal p ν gives v N (p ν ) = v β (p ν ).…”

“…Tucker introduced in [10] the notions of contibution and critical contribution of a jumping number by a divisor. The contribution of a prime divisor had earlier been defined by Smith and Thompson in [9].…”

“…The notion of contribution to a jumping number by a divisor was introduced by Smith and Thompson in [9], and the investigation has then been continued by Tucker in [10]. It turns out in Theorem 4.16 that every critically contributing divisor, in the sense of Tucker, is of the type γ∈S F E γ .…”

A formula for jumping numbers in a two-dimensional regular local ring

“…Recall that by Proposition 9 any jumping number is supported at some vertex γ with d γ > 0 or v Γ (γ) > 2. Therefore it is enough to consider the sets H a γ j for j = 1, 3,7,8,9,13,14,16,19 and 20. By Theorem 23 we know that…”

“…Besides [8], jumping numbers of simple complete ideals or analytically irreducible plane curves have been independently investigated by several people (see [12], [15] and [4]). In a local ring at a rational singularity of a complex surface, Tucker presented in [16] an algorithm to compute the set of jumping numbers of any ideal. Recently, Alberich-Carramiñana, Montaner and Dachs-Cadefau gave in [1] another algoritm for this purpose.…”