  We determine the defining equations of the Rees algebra and of the special
fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We
prove that their initial algebras are ladder determinantal rings. This allows
us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay
domains, they are Koszul, they have rational singularities in characteristic
zero and are F-rational in positive characteristic.
