Finiteness of cominuscule quantum k-theory

The product of two Schubert classes in the quantum K-theory ring of a homoge- neous space X = G=P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only nitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schu- bert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties dened by two Schubert va- rieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that whenX is cominuscule, all boundary Gromov-Witten varieties dened by three single points in X are rationally connected.