Symmetric star transforms and the algebraic geometry of their dual differential operators

Abstract

The star transform is a generalized Radon transform mapping a function on $\mathbb{R}^n$ to the function whose value at a point is the integral along a union of rays emanating from the point in a fixed set of directions, called branch vectors. We show that the injectivity and inversion properties of the star transform are connected to its dual differential operator, an object introduced in this paper. We prove that if the set of branch vectors forms a symmetric shape with respect to the action of a finite rotation group $G$, then the symbol of its dual differential operator belongs to the ring of $G$-invariant polynomials. Furthermore, we show that star transforms with degenerate symmetry correspond to linear subspaces contained in the zero set of certain elementary symmetric polynomials, and we investigate the associated real algebraic Fano varieties. In particular, non-invertible star transforms in dimension 2 correspond to certain real lines on the Cayley nodal cubic surface.