Research problems
-----------------
All the problems involve implementation (in Magma, or some other system)
of some procedure.  

1. Explicit Kustin--Miller unprojection.
The starting point is the paper
    S.\ Papadakis,
    Kustin-Miller unprojection with complexes,
    to appear in J. Algebraic Geometry
    preprint available on the web at math.AG/0111195
or the papers of Kustin and Miller in the 1980s.  The Kustin--Miller method
is to construct a large free resolution (that is, a suitable sequence of
maps between sums of polynomial rings) from two smaller ones.  One must
lift polynomial maps, a problem that can be solved with Groebner basis
techniques.  Papadakis' paper works out two substantial examples by hand.
A computer implementation would certainly find more difficult cases.

2. Recognising equations.
Given a system of polynomial equations, can you recognise 'formats'
inside them?  The point is to use as little Groebner basis as possible:
aim to work with many equations in many variables where GB is not
likely to be feasible.  Even in simple cases there can be a big payoff.
As a first sample case, consider the question: given three equations
that are the 2 x 2 minors of a 2 x 3 matrix A, can you recover A?
For example, if the equations are
	xz = y^2, xt = yz, yt = z^2
can you see
	A = ( x  y  z )
	    ( y  z  t )  ?
Since everything is in low degree, you could regard this as a problem
in linear algebra.  There are plenty of other interesting formats.

3. Nonsingularity of varieties.
This problem is not yet very well posed.  Given a family of curves or
surfaces, one has theorems that say that the typical member of the
family has good properties, like nonsingularity for example.  In practice,
it is hard to write down a member that is both general enough to have
the required property, but also special enough to be computable.  The
balance is like that between having lots of nonzero coefficients in a
polynomial (to ensure it doesn't have multiple roots), and few
nonzero coefficients (to keep it simple).
One starting place is the paper
    D.\ Eisenbud, B.\ Sturmfels,
    Finding sparse systems of parameters.
    J. Pure Appl. Algebra  94  (1994),  no. 2, 143--157.
During the lectures, we will see a number of examples where this may be
applied.

4. Models of elliptic fibrations: Ryder's algorithm.
An elliptic fibration is a surface S with a map to a curve for which the
preimage of a point is an elliptic curve.  One could also think of this
as being a particular model of an elliptic curve in the usual Weierstrass
form
	y^2 = x^3 + ax^2 + bx + c
where a, b, c are functions on a curve rather than simply numbers.
A surface may have many elliptic fibrations.  There is an unpublished
algorithm of Ryder that in a particular context (which includes the
cubic surface) computes all other models that are elliptic fibrations.
The algorithm uses torsion calculations in an elliptic curve as well
as basic polynomial calculations so is well-suited to a system like
Magma that includes Groebner basis as well as more number-theoretic
features such as analysis of elliptic curves over many fields.

5. Sarkisov program for surfaces and models of fibrations in scrolls.
This is a very well understood procedure for studying maps from the
projective plane to itself.  The traditional solution uses Cremona
transformations of the plane; more recent, but nevertheless traditional,
versions use simple transformations of rational scrolls.  A clean
implementation of the latter is possible.  The real benefit would come
if that could be extended in a limited way to higher dimensional cases.
Although very difficult mathematical issues arise then, it is still true
that somebody with a routine that could make some polynomial tricks
analogous to the surface case would be able to find applications to
finding nice models of varieties in rational scrolls.
This is also related to the paper
    J.\ Schicho
    A degree bound for the parameterization of a rational surface.
    J.\ Pure Appl. Algebra  145  (2000),  no. 1, 91--105
so would also have applications to surface parametrisation algorithms.