Algorithms in algebraic geometry by Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese

By Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese

In the decade, there was a burgeoning of task within the layout and implementation of algorithms for algebraic geometric compuation. a few of these algorithms have been initially designed for summary algebraic geometry, yet now are of curiosity to be used in functions and a few of those algorithms have been initially designed for purposes, yet now are of curiosity to be used in summary algebraic geometry.

The workshop on Algorithms in Algebraic Geometry that was once held within the framework of the IMA Annual application 12 months in functions of Algebraic Geometry by means of the Institute for arithmetic and Its purposes on September 18-22, 2006 on the college of Minnesota is one tangible indication of the curiosity. a hundred and ten members from 11 nations and twenty states got here to hear the various talks; speak about arithmetic; and pursue collaborative paintings at the many faceted difficulties and the algorithms, either symbolic and numberic, that light up them.

This quantity of articles captures a number of the spirit of the IMA workshop.

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Xy] = {[Xwo] o y = wow y=lwow,f(y)=(~)-f(w) . 3) o = (n, n - 1, . . ,1) is the longest permutation in Sn, of length dim(Fl n ), and [Xwo] is the class of a point. 3) we have Here W (~) = [Xu] . [Xv] , [Xwowl = c~,v[Xwol . 4) when f( u) + f(v) + f( W ow) = (~) and F. , G. , H. are three generally chosen flags. Note, it is not sufficient to assume the three flags are pairwise 26 SARA BILLEY AND RAVI VAKIL transverse in order to get the expected number of points in the intersection. There can be additional dependencies among the subs paces of the form Fi n Gj n Hi , The main goal of this article is to describe a method to find all flags in a general d-fold intersection of Schubert varieties when the intersection is zero-dimensional.

Washington . edu) ; supported by NSF grant DMS-9983797. edu) ; supported by NSF grant DMS-0238532. , Y. and Z•. One concrete solution to this problem, due to Lascoux and Schiitzenberger [Lascoux and Schiitzenberger, 1982], is to compute a product of Schubert polynomials and expand in the Schubert polynomial basis . The coefficient indexed by u, v , w is the solution. This corresponds to a computation in the cohomology ring of the flag variety. (Caution: this solution is known to work only in characteristic 0, due to the potential failure of the Kleiman-Bertini theorem in positive characteristic, d.

The key example: triple intersections. We now implement the algorithm of the previous section in an important special case. Our goal is to describe a method for directly identifying all flags in X = Xu (E;) n X v(E;) n Xw(E~) when l( u) + l (v) + l(w) = ( 2) and E; , E; , and E; are in general position. 4) . There are two parts to this algorithm. 1 to find the unique permutation array P c [n]4 with position vector (u, v, w) such that Pn = Tn ,3 . 10) to find all flags in X . As a demonstration, we explicitly compute the flags in X in two cases.

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