Rationality of Parameterizing Varieties for Modules Over Finite-Dimensional Algebras
One can use classical varieties to attack the problem of classifying finitely-generated modules over finite-dimensional algebras. Given such an algebra, one can write down a number of varieties which parameterize modules with certain isomorphism invariants. Furthermore, these varieties come with morphic actions by algebraic groups whose orbits are in one-to-one correspondence with isomorphism classes of such modules. Using path algebras modulo relations, we can exploit the quiver structure to learn about the structure of these varieties. We use this to give a proof of rationality of one such variety parameterizing graded modules.