Computations in algebraic geometry

Grobner basis techniques are used to obtain algorithmic solutions of some problems in polynomial rings.; Given an ideal I, a basis for I is called a homogeneous basis if homogenization of its elements generates Ih. The traditional approach for finding a homogeneous basis is to find a Grobner basis with respect to a graded order. The basis obtained by this approach is, however, generally not a minimal homogeneous basis. Buchberger's algorithm for finding Grobner basis is modified to obtain a minimal homogeneous basis.; A method is developed to find all liftings of a homogeneous ideal. As an application, the first example of a homogeneous ideal in three variables which is not liftable to a radical ideal is given. The method is extended to find the liftings of a free resolution.; Given a projective variety V⊂Pn , its ideal I(V) and a linear subspace L which does not intersect V. The projection of V to a complimentary linear subspace L' is a well-defined morphism on V. A method based on Grobner basis techniques is developed to find a basis of the ideal of the image of V under the projection provided with a basis of I(V).