For smooth projective varieties X and Y, the moduli space of degree-d algebraic morphisms from X to Y admits an algebro-geometric configuration-space model, an analogue of Bendersky-Gitler's topological result on mapping spaces. Under a point-separation (interpolation) condition on the polarization of Y, one constructs a first-quadrant spectral sequence of Galois representations / mixed Hodge structures converging to the compactly-supported cohomology of the moduli of morphisms, with terms built from ordered configuration spaces of X, the cohomology of the Picard variety, and auxiliary schemes; the model yields homological stability when the target is projective space. Students learn how configuration-space filtrations transfer from topological to algebraic function spaces.
For smooth projective varieties X and Y over an algebraically closed field of characteristic zero, the moduli space Mor_d(X,Y) parametrizes degree-d algebraic morphisms from X to Y. This paper proves…