Hypersurface Puns

Please be gentle with me. I am very much a lay person, but I would love to understand what that means. (Found it in this paper on cosmology.)

Since all nondegenrare quadratic forms over an algebraiclly closed field are isomorphic, each projective space P^n has a unique nonsingular quadratic hypersurface up to isomorphism. I was wondering if it is known what these are for each dimension.

For dimension 2, the hypersurface is just P^1 .

For dimension 3, the hypersurface is P^1 × P^1 .

I do not think it is always just going to be P^1 times itself n-1 times, as I cannot think of an embedding of P^1 × P^1 × P^1 into P^4 .

I do know that they just always be rational, but what more is known?

Having some trouble with this question: https://imgur.com/a/3QosfuS

Managed to find the simple equation f(t,x,y,z,L) = 0. And then I found dt,dx,dy,dz in terms of the new coordinates and subbed into ds^2 , but I've ended up with a dL^2 in my final expression as well as Ls everywhere. This was the final expression I got: https://imgur.com/a/C6GcquS is having L in the expression for ds acceptable, or should I have removed it all some how? Any help would be great!

I use mathematical concepts to help me make decisions, and I'm hoping to find some philosophical (or other) terminology for the same concepts.

Should I do the thing or not do the thing?

With no other information, say the answer is no.

Name one piece of relevant information and I can draw a one dimensional graph with some "yes" and some "no" on it, depending on that information. Maybe I should do the thing if today is a weekend, but not if it's a weekday.

do the thing?
Mon Tue Wed Thu Fri Sat Sun
 no  no  no  no  no yes yes

The delineation between yes and no can be described pretty concisely.

Give me a second piece of relevant information and I can draw a two dimensional graph. Maybe the younger you are, the fewer days it's acceptable to do the thing.

do the thing?
Age Mon Tue Wed Thu Fri Sat Sun
<13  no  no  no  no  no  no  no
<18  no  no  no  no  no yes yes
<21 yes  no  no  no yes yes yes
>21 yes yes yes yes yes yes yes

Notice how there's still a line between "yes" and "no", it's just not a straight line. That line can still be described.

I can't draw it in text, but if you add a third piece of relevant information then you get a 3d space with volumes of yes and no, and the surfaces between those volumes can still be described if you're patient enough.

This process can continue for arbitrarily complex scenarios. The 2-dimensional answer is an approximation of some view of the 3-dimensional answer; it might be a slice through it, or an average of each set of answers, or something else, but it will bear some resemblance to it, as will the 3 and 4 dimensional answers, and so on.

One tool that I use to find out where someone draws the point/line/surface/hypersurface in question, if they aren't aware of it themselves, is to ask them to describe two situations that are as close together as possible but with different answers. Those two situations are two points on this graph, on opposite sides of the line. With enough of those pairs of points, the shape of the line can be narrowed down.

Having described all of that... Is there some non-mathematical vocabulary that would help me find information on this subject, or to discuss it?

Because she had read that the region Ω would then contain a worm hole!

(x-post r/MoleJokes)

I'm reading Katz "Enumerative Geometry and String Theory".

He shows that a degree d hypersurface has cohomology class dH (where H is the class of a hyperplane) by the following argument (p.80, I'm paraphrasing a bit):

...we can continuously deform a degree d hypersurface (given by F=0) to a union of d hyperplanes by the equations:

[; G_t := tF(x) + (1-t) \Pi_{i=1}^{d} l_i (x) = 0 ;]

where [; l_i ;] are homogeneous linear forms.

I've studied the topic of intersection theory a bit already, so my question isn't about the result necessarily, but more about his argument. Specifically, what does he mean by continuous here (i.e. continous in what topology)?

For example, if we are in the the plane, and we take d=2, he is saying we can continuously deform a circle into a pair of lines. I'm pretty sure that deformation isn't continuous, though (or maybe it is in the complex plane?)

shit sucks

I guess the concept didn't work

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Because she had read that the region Ω would then contain a wormhole.