Wikipedia:Reference desk/Archives/Mathematics/2016 October 21

= October 21 =

The union of lines joining points on a variety to a fixed point on it
Let $$Y \subset \mathbb P^n$$ be a projective variety, and fix a point $$p \in Y$$. Define X to be the closure of the union of all lines (pq) where $$q \in Y \setminus \{p\}$$. Assume $$Y \neq X$$. How can I find a birational morphism $$Y \times \mathbb P^1 \rightarrow X$$? trying to parameterize the line (pq) by P^1 and send (q,t) to the point corresponding to the parameter t is not one-to-one. This is exercise I7.7 (a) in Hartshorne.--46.117.104.173 (talk) 09:56, 21 October 2016 (UTC)


 * I don't think in general there is a birational map between these two varieties.  Sławomir Biały  (talk) 10:18, 21 October 2016 (UTC)
 * The exercise asks to prove dim(X)=dim(Y)+1, and an online hint is to find a birational map $$Y \times \mathbb A^1 \rightarrow X$$. I believe this to be equivalent to the existence of a (perhaps more natural) birational map $$Y \times \mathbb P^1 \rightarrow X$$, since $$Y \times \mathbb A^1$$ is open in $$Y \times \mathbb P^1$$.--46.117.104.173 (talk) 11:14, 21 October 2016 (UTC)
 * You don't actually need a birational map to prove that the dimension is the same. The "obvious" mapping from $$Y\times\mathbb A^1\to X$$ is not birational when Y is a hypersurface, but it generically has maximal rank.  You can use this to give affine charts on X.   Sławomir Biały  (talk) 13:57, 21 October 2016 (UTC)
 * I think that a truly birational map can be defined by taking an affine set $$U_i$$ containing the point p, and send (q,t) to the point on the affine line (pq) that corresponds to the parameter t. Then if the line (pq) intersects Y in an additional point r, the map can't decide which parameter to use, so we take the parameterization with unit speed $$(q,t) \mapsto p+t\frac{(q-p)}{d(p,q)}$$. Is that all right?--46.117.104.173 (talk) 21:52, 25 October 2016 (UTC)

Rare math symbol unicode
I am trying to quote an old mathematics book. It is using a three-dot notation that I've never seen, so I don't know the name, so I cannot find the unicode for it. I need the unicode so I can simply reprint the original. The two symbols used have three dots. One symbol has a high dot on the left, a mid-dot in the middle and a mid-dot on the right. The other symbol has a low dot on the left, a mid-dot in the middle, and a mid-dot on the right. I've been searching through unicode charts, but I don't know how to efficiently find rarely used symbols. 209.149.113.4 (talk) 13:09, 21 October 2016 (UTC)
 * Try http://shapecatcher.com. PrimeHunter (talk) 13:14, 21 October 2016 (UTC)


 * Thanks. That helped me realize that I could use Braille to fake it. 209.149.113.4 (talk) 13:55, 21 October 2016 (UTC)


 * I tried shapecatcher and it said it was "Down right diagonal ellipsis, Unicode hexadecimal: 0x22f1, In block: Mathematical Operators". That's &amp;#8945; = &#8945; So I'm duly impressed... especially considering the quality of my scrawled circles and the fact that I don't think I've ever seen this symbol before.  Oh, and as you might have guessed, the other is next to it at &amp;#8944; = &#8944; Wnt (talk) 12:58, 22 October 2016 (UTC)


 * But the OP's descriptions have two dots on the same level. —Tamfang (talk) 17:43, 22 October 2016 (UTC)


 * Erm.... ooops! (This kind of thing happens way too often when I start playing with a new tool... too busy admiring it to remember what I was trying to do) Wnt (talk) 11:22, 24 October 2016 (UTC)