Anisotropic Discriminants

Abstract

Let F ₁,…, F _n be homogeneous polynomials of positive degrees in the polynomial algebra A[T] = A[T ₀,…, T _n ] graded by arbitrary positive integral weights for the indeterminates such that D: = A[T]/(F ₁,…, F _n ) is a complete intersection of relative dimension 1 over the commutative noetherian ring A. Then Proj D is an affine scheme over A and its algebra B: = Γ(Proj D) of global sections is finite and stably free. A formula for the discriminant d _B|A  = Discr(F ₁,…, F _n ) of B over A is given generalizing the well-known formula for the discriminant of an A-algebra of type A[X]/(G), where G is a monic polynomial in one variable. The formula is a special case of a result on discriminants for A-bilinear forms on B derived from linear forms B → A. The general formula uses the description of the linear forms on B with the help of a duality theory for D and the theory of resultants.

Key Words:

• Anisotropic discriminant
• Discriminant
• Duality theory
• Graded complete intersection
• Resultant

2000 Mathematics Subject Classification:

• 13C40
• 13A02
• 13E10
• 14M10

Notes

Communicated by W. Bruns.