Research Statement on Mathematics

Thesis Work on Algebraic Geometry The homogeneous cubic polynomials in four variables x, y, z and w form a 20 dimensional vector space over C. Let S ⊂ P be a surface defined by the equation f(x, y, z, w) = a0x 3 + a1x y + · · ·+ a19w = 0, which naturally determines a point [a0, . . . , a19] ∈ P. X admits a natural G = SL(4;C) action. Let X denote the smooth cubic surfaces, X the semistable ones and X the stable ones. Two cubic surfaces are isomorphic if one can be transformed to the other by some element of SL(4;C). Let R = C[a0, . . . , a19] be the invariant graded subring, which is finitely generated, and let M = ProjR a projective variety. GIT yields a surjective morphism X →M. Let X denote the isomorphism classes of smooth cubic surfaces and M the isomorphism classes of stable cubic surfaces, i.e, the cubic surfaces with at worst nodal singularities locally isomorphic to x + y = z. The bad news is that M does not have a moduli interpretation, i.e, as the isomorphism classes of cubic surfaces with prescribed singularities. In the thesis, I give a geometric compactification of Ms.