Timeline for How can one compute the canonical class of the projective completion of the tautological bundle over $P^1\times P^1$?

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Aug 24, 2011 at 12:38	comment	added	Allen Knutson		Oh whoops, you're right about the signs of course. Fixed.
Aug 24, 2011 at 12:37	history	edited	Allen Knutson	CC BY-SA 3.0	added 72 characters in body
Aug 24, 2011 at 5:59	comment	added	Dhruv		Thats a great response, thank you sir. I wanted to understand it free from the toric machinery via something like Leray-Hirsch. But thanks again, largely for the sake of insight, but thanks nonetheless. One comment I will make, I think on P1xP1 if you take the four edges of the polytope clockwise I believe you should get h1, h2, h1, h2 (no minus signs). As a quick check, the anticanonical class of P1xP1 is 2(h1+h2). Doesnt change the answer though.
Aug 24, 2011 at 4:39	comment	added	Allen Knutson		I find the adjunction formula easiest to remember in terms of the rule in Step 4. Let $\partial X$ denote the anticanonical class of $X$. Then adjunction in general says that if $\partial X = [D \cup E]$, then $\partial D = [D \cap E]$. Now picture $X$ as a polytope, $\partial X$ as its boundary, $D$ as one facet, and $E$ as the rest of the facets (a hemisphere decomposition, topologically). This says that $D\cap E$ is the boundary of $D$.
Aug 24, 2011 at 4:34	history	answered	Allen Knutson	CC BY-SA 3.0