I know the Milnor and Tjurina numbers reflect in certain way the complexity of
a singular point for a variety defined by one polynomial V=v()).
Now I know there exists some generalizations (or at least attempts have been made to generalize) similar quantities for a variety defined by more than one polynomial V(f_1,...,f_n).
For example the Milnor number should generalize to object called 'Milnor Class' in Chow's group.
I have mainly 2 questions regarding these possible generalizations:
1) Does anybody know a good reference book in which such problems would have been investigated, discussed and possibly rigorously defined.
2) Are there any known algorithms to compute such objects ? And if so does anyone know an article or a book where such algorithms could have been described ? Or at least partial results ?
From the little that I know about the subject the proper generalisations come from the theory of mixed Hodge structures. I have been told that a good place to start is the book by Eduard Looijnga
Looijenga, Eduard. Isolated singular points on complete intersections. Vol. 77. Cambridge University Press, 1984. http://intlpress.com/site/pub/files/preview/bookpubs/00000407.pdf
Looijenga, Eduard, and Joseph Steenbrink. "Milnor number and Tjurina number of complete intersections." Mathematische Annalen 271.1 (1985): 121-124.
IIRC Milnor's little book on hypersurface singularities already contains several definitions of the Milnor number.
The SINGULAR system (available through SAGE) is heavily into computing invariants of singularities.
Hi Rogier
Thank you for very much for your answer and book references are always appreciated! Yes I know these are implemented somehow in Singular. In fact Singular is one the programs that I mainly use. Unfortunately the cases we are dealing with are not always complete intersections, although in most cases they are and are generated by codim number of generators.
Also are you aware about the Hilbert-Samuel function/multiplicity number for singularities ? Do you remember any good books where the algorithms and def.s for these would be introduced in . I would really appreciate all book tips.
Also about a question about the 'intersection multiplicity': If two varieties intersect
transversally do you remember if there are kind of 'straight' or sequence of commands to verify this with Sage, Singular, Macaulay2 or CoCoa without coding the algorithms completely yourself ? Just to save time.
Also if you remember good books where these problems are tackled it
would also be good to know !
But thanks again for you response !
regards
Samuli Piipponen
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