These scripts are designed to analyse surfaces in P4.
	- smoothness
	- few_smooth
	- speedy_smooth
are scripts, which check the smoothness. The first uses the staight forward
method based on the implicit function theorem. The two other sripts try
to use less memory and time. The most expensiv part of the computation by
the straight forward method is to compute all minors of the jacobian matrix.
Few_smooth just picks a few of the minors and hopes for the best. It assumes
that the ideal is generated anly in 2 different degrees. 
Speedy_smooth is a more sophisticated method. It assumes that the smallest
degree generators of the ideal are quintics. Then if a choosen quintic generator
has only isolated singularities, and the degree of the singular set coincides
with the degree of the zero loci of the corresponding section of the normal
bundle of the surface, and this is the expected degree, then one can deduce the
smoothness if the variety defined by the quintics has embedding dimension at
most 3 at all points of the surface. All this can be checked by less expensive
computations than the computation of all minors.
	The other sripts are for the adjunction process of the surface.
	- first_adjoint 
computes the image of the surface under H+K, and the exceptional lines, or
rather the ideal of the corresponding points in the image surface.
	- next_adjoint
computes the second or higher adjoint surfaces, and hence exceptional conics,
twisted cubics etc of the original surface. It assumes that the surface
is arithmetically Cohen_Macaulay and three-regular in sense of Castelnuovo-
Mumford, which is reasonable to expect from an adjoint regular surface. 
Also in both scripts pg=0 is assumed, which however could be overcomed by a
slight modification of the program.
	- dualizing
computes a presentation of the dualizing module under the assumption that
the surface is arithmetically Cohen-Maculay, 3-regular and that
the last variables form a regular sequence. It is used by next-adjoint.
	- check_para_sys
checks whether the last variables form indeed a regular system of parameters,
	- random_aut
performs a random linear coordinate transformation. It is used by next_adjoint
in the attemp to obtain that the last variables form a regular system.
The scripts
	- xring
	- put
	- get
are little things I find convenient to use,
	- ffrobenius
works for small characterics, and gives the fix points of the Frobenius 
endomorphism and it square.
	For comments and questions on the scripts please contact me under
		schreyer@btm8x2.mat.uni-bayreuth.de


