Hello everyone,


I'm trying to look at Mumford's Paper, [The Pathologies of Modular Surfaces](http://www.dam.brown.edu/people/mumford/Papers/DigitizedAlgGeomPapers--ForNon-CommercialUse/61b--Path1.pdf).


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!



Question about Terminology in Mumford
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.


