From: "Robert L. Bryant" <bryant@math.duke.e.du>
Newsgroups: sci.math.research
Subject: Re: cubic surface classification
Date: Wed, 19 Aug 1998 09:10:06 -0400

You wrote:

>I have a question in algebraic geometry:
>given q, a polynom of degree 3 over P^3 (The Projective space) , is
>there a way to know whether it is reducible, based on its coefficients ?
>is there something similar to quatratic polyoms classification, based on
>the coefficients matrix determinants etc. ?
>i.e. if q defined by a contraction of 3 vectors [x,y,z,w] with a
>symetric trilinear tensor of dimension 4x4x4, what are the conditions on
>a tensor defining a reducible polynom.
>if anyone know a good reference to this question, based on simple
>algebraic geometry, it can be very helpful

	Well, I was hoping that an algebraic geometer would answer
this question for us, but since no one has, I'll say what I know.
Maybe this will prompt a response from someone who really knows the
answer.

	The homogeneous polynomials of degree 3 in 4 variables form
a vector space of dimension 20, so the associated projective space
is a P^{19}.  The reducible ones are the image of the natural map
B: P^3 x P^{9}  -->  P^{19} that is induced by the bilinear map without
zero divisors  C^4 x C^{10} --> C^{20} that takes a pair (l,q)
consisting
of a linear polynomial l and a quadratic polynomial q (each in 4
variables)
and sends them to their product lq.  The image B(P^3 x P^{9}) is an
algebraic variety of dimension 12 and degree 220 ( = {9+3\choose3} ).
Thus, by Chow's Theorem, this image is defined by the ideal I of 
polynomials that vanish on it.  This answers your question in the 
affirmative, at least abstractly.  Presumably, though, you will want 
to construct generators for this ideal and, for that, you'll need
to know something about the Hilbert polynomial of the ideal I, probably.
Since the image is of codimension 7 in  P^{19}, the ideal I will have
to have at least 7 generators.  If it did have 7 generators (so that
the variety is what is known as a complete intersection), then they
would have to have an average degree of more than 30.  (A polynomial
of degree 30 in 20 variables is an awesome thing to contemplate.)
Thus, it strikes me as unlikely that you really want to test
cubic polynomials for factors by this method.

	It's instructive to consider this problem in three variables.
Then the relevant mapping is B: P^2 x P^5  -->  P^9, the image
is of dimension 7 and degree 21 ( = {5+2\choose2} ).  If this
were a complete intersection, it would have to be defined by the
vanishing of two polynomials, one of degree 3 and another of degree 7.
(The other possibility would be one of degree 1 and the other of
degree 21, but the reducible cubics clearly do not lie in any
linear subspace of codimension 1.)  However, it is easy to see that
there is no SL(3,C)-invariant polynomial of degree 3 on S^3(C^3).
Thus, the image B(P^2 x P^5) \subset P^9 cannot be a complete
intersection.  I have no guess about the structure of the ideal
defining this image variety, but I'm sure that this is a well-studied
classical subject.  Can some algebraic geometer help us?

	Yours,

	Robert Bryant