Failure of Hasse principle for intersections of two quadrics

The Minkowski–Hasse theorem states that if a quadratic form with rational coefficients represents zero in in all completions of $\mathbb{Q}$ , then it represents zero over $\mathbb{Q}$. I will show that an analogous statement for a system of two quadric forms in five variables is false and give a geometric interpretation of the obtained fact. The talk is based on a paper by V.A.Iskovskikh.