Timeline for Intuition behind generic points in a scheme
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2012 at 19:02 | answer | added | Heinrich Hartmann | timeline score: 10 | |
| Feb 24, 2012 at 18:22 | comment | added | quim | @a-fortiori: actually (2) always implies (1) when a generic point exists. @ssquid: Your statements are true for 'general' points (over a nonempty Zariski open) so they fail only in a closed proper subset. The generic point belongs to no proper closed subset. The first example, for instance, means that if you take the degree m generic curve, ie the curve whose coefficients are algebraically independent (variables) over the base field k $a_1,\dots,a_N$, then there are nm distinct intersection points (whose coordinates are algebraic over k(a1,…,aN)). | |
| Feb 24, 2012 at 17:09 | comment | added | Donu Arapura | I decided to delete my answer since it may not be getting to the heart of you are after. Although, and I don't mean to be impolite, I'm not quite sure what that is. | |
| Feb 24, 2012 at 16:17 | history | edited | ssquidd | CC BY-SA 3.0 |
added 189 characters in body
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| Feb 24, 2012 at 16:12 | comment | added | ssquidd | That is indeed the point of view I would like to explore. Could you give me more detail? How do you form such a "scheme of possible polynomials"? I can see that a degree-$m$ polynomial is determined by a list of coefficients, and the space of such polynomials actually form a projective space since scaling of coefficients does not change the polynomial equation. How does this connect to the scheme you were talking about? Thanks. (P.S., your comment is more like an answer than a comment) | |
| Feb 24, 2012 at 15:06 | answer | added | Liviu Nicolaescu | timeline score: 2 | |
| Feb 24, 2012 at 6:42 | comment | added | Will Sawin | I'm not sure you actually will get a nice answer that looks like $n\cdot m$ points or the cone on it, though. | |
| Feb 24, 2012 at 6:41 | comment | added | Will Sawin | To apply generic points to the first one, we take the ring of projective space, or at least the cone on it: $\mathbb C[x,y,z]$, and then quotient out by a degree-$n$ and degree-$m$ homogeneous polynomial. What are the coefficients? The degree-$n$ polynomial is any one you specify, but the coefficients of the degree $m$ polynomial are new invariants. The scheme of possible polynomials is an affine space. The generic point corresponds to looking at things over the field generated by those invariants. | |
| Feb 24, 2012 at 6:30 | comment | added | user2035 | There are two meanings of a property being generically true: (1) over the generic point and (2) over a nonempty open subset. One important thing about generic points is that in many cases (1) implies (2). See also mathoverflow.net/questions/28496/… | |
| Feb 24, 2012 at 5:39 | history | asked | ssquidd | CC BY-SA 3.0 |