Cremona transformations and derived equivalences of K3 surfaces

Two varieties are called derived equivalent if their bounded derived categories of coherent sheaves are equivalent to each other. In the case of K3 surfaces, this equivalence is realized as an Hodge isometry between the transcendental lattices according to Mukai and Orlov. Could it be realized further through an explicit construction of birational geometry? In this talk, I will present an example where the derived equivalences of K3 surfaces are explained through Cremona transformations of ${\bf P}^4$. This example also provides an interesting relation in the Grothendieck ring of complex algebraic varieties. This is joint work with Brendan Hassett.