# What is known about the “moduli space of morphisms” $X \to Y$?

Let $X$ and $Y$ be projective varieties. I am assuming that there is some construction of a "moduli space" parametrizing the morphisms $X \to Y$. For instance, if one identifies such morphisms with their graphs in $X \times Y$, one can at least hope that these graphs would correspond to a locally closed subscheme of the Chow or Hilbert schemes for $X \times Y$, although this may not be the best way to construct such a space. In the case of $X = Y = \mathbb{P}^n$, this space should probably be a disjoint union of $PGL_n$ with $\mathbb{P}^n$ [Edit:This is not correct; I know a theorem that the image must be either a point or all $\mathbb{P}^n$, but grossly misapplied it by neglecting maps of degree > 1]. (I am requiring the varieties to be projective because, for instance, if $Y$ is the affine line, then the morphisms $X$ to $Y$ would naturally correspond to the global sections of $X$, which for general $X$ is too big to fit in one nice scheme, being infinite-dimensional. Although I suppose it might work if we confined ourselves to e.g. morphisms of a fixed degree, and we are already doing something of this sort in looking at Chow or Hilbert schemes, so perhaps my concern here is needless.)

Is there a standard space that represents these morphisms, and if so, what is known about it? Is there a good reference for this?

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See the book by Koll\'ar, "Rational Curves on Algebraic Varieties", Chapter 1. –  mdeland Nov 4 '10 at 17:27
Also, to address your example when $n = 1$, degree $d > 0$ maps from P^1 to P^1 may be identified with two degree $d$ homogeneous polynomials in two variables with no common factor (up to a total scaling). So the Hom scheme in this case is much bigger than you suggest! –  mdeland Nov 5 '10 at 1:03
mdeland: Thanks for the correction. I don't know what I was thinking. –  Charles Staats Nov 5 '10 at 2:20

It is called the Hom scheme, and I think it's defined in Grothendieck's Bourbaki 221 paper, where he constructs the Hilbert and Quot schemes. For any $S$-schemes $X$ and $Y$, $\underline{Hom}_S(X,Y)$ assigns to any $S$-scheme $T$ the set $Hom_T(X_T,Y_T)$. Osserman has a quick overview of the constructions, which you can find with Google. Olsson's paper "Hom stacks and restriction of scalars" is one possible reference that covers a more general setting.