Question

Hello everyone,

I'm trying to look at Mumford's Paper, The Pathologies of Modular Surfaces.

On page 341, section II he says a certain surface can be constructed as the join of 3 graphs $E_0 \rightarrow \mathbb{P}^1$. What is the join of graphs?

Cheers!

Answer 1

The join of two varieties $X,Y\subseteq \mathbb{P}^n$ is $$ J(X,Y) = \overline{\bigcup_{\substack{x\in X,~y\in Y\\x\ne y}} \ell(x,y)}$$ where $\ell(x,y)$ denotes the projective line through $x$ and $y$. The join of $k$ varieties $X_1,\ldots,X_k\subseteq \mathbb{P}^n$ is defined to be the closure of the union of the corresponding, projective $(k-1)$-folds, or by induction $$J(X_1,\ldots,X_k) := J(X_1,J(X_2,\ldots,X_k))$$

This definition is from Joseph Landsberg's book Tensors: Geometry and Applications, page 118. The graph of a regular function is a projective variety, so this should be defined.

Answer 2

A reference for the join of projective varieties, in Grothendieck's EGA style is

Altman, Allen B.; Kleiman, Steven L.: Joins of schemes, linear projections. Compositio Mathematica, 31 no. 3 (1975), pp. 309-343

You can dowload it here.