Simmetrioiden

Simmetrioiden, or symmetroids, are a class of determinantal hypersurfaces studied in algebraic geometry. In a projective space P^n over a field of characteristic not equal to 2, fix a symmetric m×m matrix A_i with entries in the base field, for i = 0,…,n, and form the linear combination A(x) = x0 A0 + x1 A1 + … + xn An, whose entries are linear forms in the homogeneous coordinates x = [x0:…:xn]. The determinant det A(x) is a homogeneous polynomial of degree m in x, and the zero locus S = {x ∈ P^n | det A(x) = 0} is called a symmetroid.

Basic properties: S is a hypersurface of degree m in P^n. For generic choices of the Ai, S is irreducible; its singularities occur at points x where the rank of A(x) drops below m−1, in particular where rank ≤ m−2. Thus the singular locus reflects degeneracies of the family A(x).

Examples and special cases: When n = 3 and m = 4, S is a quartic surface in P^3 known as a quartic symmetroid. A generic quartic symmetroid has 16 ordinary double points as singularities. The double cover of P^3 branched along S, after resolving singularities, is a K3 surface; this provides a classical link between symmetroids and K3 geometry.

Relation to discriminants and quadrics: S can be viewed as the discriminant locus of the linear system of quadrics {Q_x(y) = y^T A(x) y}. It encodes the parameter values for which Q_x becomes singular. The geometry of S is governed by the rank stratification of the matrix family A(x).

Further study: Symmetroids appear in determinantal varieties, moduli of K3 surfaces, and explicit constructions in projective and complex geometry.