We show that every complete intersection defined by Laurent polynomials in an algebraic torus is isomorphic to a complete intersection defined by master functions in the complement of a hyperplane arrangement, and vice versa. We call systems defining such isomorphic schemes Gale dual systems because the exponents of the monomials in the polynomials annihilate the weights of the master functions...

We propose a new definition of the elliptic genera for complete intersections, not necessarily nonsingular, in projective spaces. We also prove they coincide with the expressions obtained from Landau-Ginzburg model by an elementary argument.

Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p ≤ dim C X we have a sheaf Ω p (E) whose sections are holomorphic (p, 0)-forms with coefficients in E. We set and we define the holomorphic Euler characteristics χ p (X, E) := q≥0 (−1) q h p,q (X, E). It is convenient to introduce the generating function of these numbers χ y (X, E) := p≥0 y p ...

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

We ask when certain complete intersections of codimension r can lie on a generic hypersurface in Pn. We give a complete answer to this question when 2r ≤ n + 2 in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection.

The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees generators defining ideal. are not, in general, — but sometimes they are. When? We give some general criteria and completely answer question up to codimension 6.