Question about Terminology in Mumford - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:04:17Z http://mathoverflow.net/feeds/question/103130 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103130/question-about-terminology-in-mumford Question about Terminology in Mumford uncookedfalcon 2012-07-25T22:50:35Z 2012-07-26T10:56:35Z <p>Hello everyone,</p> <p>I'm trying to look at Mumford's Paper, <a href="http://www.dam.brown.edu/people/mumford/Papers/DigitizedAlgGeomPapers--ForNon-CommercialUse/61b--Path1.pdf" rel="nofollow">The Pathologies of Modular Surfaces</a>.</p> <p>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?</p> <p>Cheers!</p> http://mathoverflow.net/questions/103130/question-about-terminology-in-mumford/103167#103167 Answer by Jesko HÃ¼ttenhain for Question about Terminology in Mumford Jesko HÃ¼ttenhain 2012-07-26T07:33:31Z 2012-07-26T07:38:43Z <p>The <b>join</b> of two varieties $X,Y\subseteq \mathbb{P}^n$ is <code>$$J(X,Y) = \overline{\bigcup_{\substack{x\in X,~y\in Y\\x\ne y}} \ell(x,y)}$$</code> 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 <code>$$J(X_1,\ldots,X_k) := J(X_1,J(X_2,\ldots,X_k))$$</code></p> <p>This definition is from Joseph Landsberg's book <em>Tensors: Geometry and Applications</em>, page 118. The graph of a regular function is a projective variety, so this should be defined.</p> http://mathoverflow.net/questions/103130/question-about-terminology-in-mumford/103183#103183 Answer by Leo Alonso for Question about Terminology in Mumford Leo Alonso 2012-07-26T10:56:35Z 2012-07-26T10:56:35Z <p>A reference for the join of projective varieties, in Grothendieck's EGA style is</p> <p>Altman, Allen B.; Kleiman, Steven L.: Joins of schemes, linear projections. Compositio Mathematica, <strong>31</strong> no. 3 (1975), pp. 309-343 </p> <p>You can dowload it <a href="http://www.numdam.org/item?id=CM_1975__31_3_309_0" rel="nofollow">here</a>.</p>