  This paper deals with properties of the algebraic variety defined as the set
of zeros of a "typical" sequence of polynomials. We consider various types of
"nice" varieties: set-theoretic and ideal-theoretic complete intersections,
absolutely irreducible ones, and nonsingular ones. For these types, we present
a nonzero "obstruction" polynomial of explicitly bounded degree in the
coefficients of the sequence that vanishes if its variety is not of the type.
Over finite fields, this yields bounds on the number of such sequences. We also
show that most sequences (of at least two polynomials) define a degenerate
variety, namely an absolutely irreducible nonsingular hypersurface in some
linear projective subspace.
