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

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October 21

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The union of lines joining points on a variety to a fixed point on it

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Let   be a projective variety, and fix a point  . Define X to be the closure of the union of all lines (pq) where  . Assume  . How can I find a birational morphism  ? 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)[reply]

I don't think in general there is a birational map between these two varieties. Sławomir Biały (talk) 10:18, 21 October 2016 (UTC)[reply]
The exercise asks to prove dim(X)=dim(Y)+1, and an online hint is to find a birational map  . I believe this to be equivalent to the existence of a (perhaps more natural) birational map  , since   is open in  .--46.117.104.173 (talk) 11:14, 21 October 2016 (UTC)[reply]
You don't actually need a birational map to prove that the dimension is the same. The "obvious" mapping from   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)[reply]
I think that a truly birational map can be defined by taking an affine set   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  . Is that all right?--46.117.104.173 (talk) 21:52, 25 October 2016 (UTC)[reply]

Rare math symbol unicode

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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)[reply]

Try http://shapecatcher.com. PrimeHunter (talk) 13:14, 21 October 2016 (UTC)[reply]
Thanks. That helped me realize that I could use Braille to fake it. 209.149.113.4 (talk) 13:55, 21 October 2016 (UTC)[reply]
I tried shapecatcher and it said it was "Down right diagonal ellipsis, Unicode hexadecimal: 0x22f1, In block: Mathematical Operators". That's &#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 &#8944; = ⋰ Wnt (talk) 12:58, 22 October 2016 (UTC)[reply]
But the OP's descriptions have two dots on the same level. —Tamfang (talk) 17:43, 22 October 2016 (UTC)[reply]
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)[reply]