Cubic surfaces and q-numerical ranges

Let A be an n × n complex matrix and 0 ≤ q ≤ 1. The boundary of the q-numerical range of A is the orthogonal projection of a hypersurface defined by the dual surface of the homogeneous polynomial F (t, x, y, z) = det(t In + x(A + A )/2 + y(A−A)/(2i) + z AA). We construct different types of cubic surfaces SF corresponding to the homogeneous polynomial F (t, x, y, z) induced by some 3 × 3 matrices. The degree of the boundary of the Davis-Wielandt shell of a 3 × 3 upper triangular matrix is determined by the cubic surface SF . AMS subject classifications: 14J17, 15A60