NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/symbolic/2352

Text File | 1992-09-08 | 1.4 KB | 34 lines

Newsgroups: sci.math.symbolic
Path: sparky!uunet!spool.mu.edu!yale.edu!think.com!mintaka.lcs.mit.edu!zurich.ai.mit.edu!ara
From: [e-mail] (Allan Adler)
Subject: vector bundles with macaulay
Message-ID: <[e-mail]>
Sender: [e-mail]
Organization: M.I.T. Artificial Intelligence Lab.
Distribution: sci
Date: Wed, 9 Sep 1992 09:55:24 GMT
Lines: 22



Suppose X is a projective variety, defined by a homogeneous ideal I in
a polynomial ring k[x0,x1,...,xn]. Suppose I have an explicit mapping
f from X into a Grassmann variety, say, the variety Gr(r,m) of
r-dimensional subspaces of a vector space of dimension m. There is a
canonical r-dimensional vector bundle E on Gr(r,m) whose fibre at
a point p of Gr(r,m) is the r-dimensional space represented by p.
Let us denote the dual of E by E'. Using f, we can pull back E and E'
to X to get vector bundles over X. One can get more vector bundles
by applying other functors, such as tensor products, exterior powers, etc.
There are also other bundles that arise naturally, such as the bundle of
m-r dimensional quotient spaces, the tangent bundle of Gr(r,m), etc,
that one can also pull back or plug into functors or both.

How can I use Macaulay to compute the cohomology of X with coefficients
in these vector bundles? In particular, how can I use Macaulay to
compute the spaces of sections of these bundles? Examples will be especially
welcome.

Allan Adler
[e-mail]