A note on the Intersection of Veronese Surfaces

Motivated by our study (elsewhere) of linear syzygies of homogeneous ideals generated by quadrics and their restrictions to subvarieties of the ambient projective space, we investigate in this note possible zero-dimensional intersections of two Veronese surfaces in P^5. The case of two Veronese surfaces in P^5 meeting in 10 simple points appears also in work of Coble, Conner and Reye in relation to the 10 nodes of a quartic symmetroid in P^3, and we provide here a modern account for some of their results.