Moduli of Cubic fourfolds and reducible OADP surfaces

In this thesis, we explore the intersection of the Hassett divisor C8, parametrizing smooth cubic fourfolds X containing a plane P with other divisors Ci . Notably we study the irreducible components of the intersections with C12 and C20. These two divisors generically parametrize cubics containing a smooth cubic scroll, and cubics containing a smooth Veronese surface respectively. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of P with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection from P, or by finding examples of reducible one-apparent-double-point surfaces inside X. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.