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

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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 ⋱ = ⋱ 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 ⋰ = ⋰ 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)