In this paper we study 15 complete intersection K3-fibered Calabi–Yau variety types in biprojective space P1 × P These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d, 0) for every positive integer d on a general Calabi– Yau variety of these types. The proof depends heavily on existence theorems for...

We prove analogs of results of Cox and Cox/Schenk on the structure of certain ideals in the bigraded polynomial ring k[s, u; t, v].

Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field K are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose endomorphism rings are orders in K. For real quadratic fields, a similar description is not known. However, the relevant (still unproved) case of Stark conjectures ([St...