# Singular

### D.5.5 resolve_lib

Library:
resolve.lib
Purpose:
Resolution of singularities (Desingularization) Algorithm of Villamayor
Authors:
A. Fruehbis-Krueger, anne@mathematik.uni-kl.de,
G. Pfister, pfister@mathematik.uni-kl.de

References:
[1] J.Kollar: Lectures on Resolution of Singularities, Princeton University Press (2007)
(contains large overview over various known methods for curves and surfaces as well as
a detailed description of the approach in the general case)
[2] A.Bravo, S.Encinas, O.Villamayor: A Simplified Proof of Desingularisation and
Applications, Rev. Math. Iberoamericana 21 (2005), 349-458
(description of the algorithmic proof of desingularization in characteristic zero which underlies this implementation)
[3] A.Fruehbis-Krueger: Computational Aspects of Singularities, in J.-P. Brasselet, J.Damon et al.: Singularities in Geometry and Topology, World Scientific Publishing, 253--327 (2007)
(chapter 4 contains a detailed discussion on algorithmic desingularization and efficiency aspects thereof)

Procedures:

 D.5.5.1 blowUp computes the blowing up of the variety V(J) (considered as embedded in V(W)) in the (smooth) center V(C), D.5.5.2 blowUp2 computes the blowing up of the variety V(J) in the (possibly singular) center V(C) D.5.5.3 Center computes 'Villamayor'-center for blow up D.5.5.4 resolve computes the desingularization of the variety V(J) D.5.5.5 showBO prints the content of a BO in more human readable form D.5.5.6 presentTree prints the final charts in more human readable form D.5.5.7 showDataTypes prints help text for output data types D.5.5.8 blowUpBO computes the blowing up of the variety V(BO[1]) in the center V(C). BO is a list (basic object), C is an ideal D.5.5.9 createBO creates basic object from input data D.5.5.10 CenterBO computes the center for the next blow-up of the given basic object D.5.5.11 Delta apply the Delta-operator of [Bravo,Encinas,Villamayor] D.5.5.12 DeltaList list of results of Delta^0 to Delta^bmax