We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with referen...

The quadratic Veronese embedding ρ maps the point set P of PG(n, F ) into the point set of PG( (n+2 2 ) − 1, F ) (F a commutative field) and has the following well-known property: If M ⊂ P, then the intersection of all quadrics containing M is the inverse image of the linear closure of Mρ. In other words, ρ transforms the closure from quadratic into linear. In this paper we use this property to...