# Question about Terminology in Mumford

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!

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 $$J(X_1,\ldots,X_k) := J(X_1,J(X_2,\ldots,X_k))$$