Timeline for The space of valuations of a function field
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14 events
| when toggle format | what | by | license | comment | |
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| Sep 23, 2011 at 10:49 | comment | added | Xin Nie | Thank you! Concerning the fifth remark, can we say that an irreducible curve passing through $p$ is completely determined by its intersection with the exceptional divisor every time when we blow up? | |
| Sep 23, 2011 at 10:19 | vote | accept | Xin Nie | ||
| Sep 20, 2011 at 14:29 | answer | added | Pedro Fortuny | timeline score: 3 | |
| Sep 20, 2011 at 14:28 | comment | added | quim | The details and classification of valuations on surfaces according to their sequences of centers is written in Casas-Alvero, "Singularities of plane curves", Chapter8. BTW, I assumed in the "coordinates" part that the surface is smooth at $p$, of course this is not really restrictive. | |
| Sep 20, 2011 at 14:27 | comment | added | quim | If you start with an arbitrary valuation on a surface, either its center is an irreducible curve, and then you have a discrete rank one valuation, or it is a point. In the second case, blow up the point to get a new model and look for the center there. If it is an irreducible curve, then you have a discrete rank one valuation, otherwise it is a point. Iterating, either you end up with a curve center (so called "divisorial valuations") or you get an infinite sequence of points, determining a point in the Zariski-Riemann space. | |
| Sep 20, 2011 at 14:23 | comment | added | quim | Fifth and last. I assume you know what the center of a valuation is in Zariski-Samuel. In this case, the center is (the maximal ideal of) the point $p$. Blow up this point on the surface $S$, obtaining a new model $\tilde S$ for the same field, and let $\tilde C$ be the strict (birational) transform of $C$ on $\tilde S$. Because $C$ is unibranch at $p$, there is a unique point $\tilde p$ on $\tilde C$ whose image by the blowup is $p$ (i.e., infinitely near to $p$). It turns out that (the maximal ideal of) $\tilde p$ is the center in this model of the valuation defined above. Iterate. | |
| Sep 20, 2011 at 14:13 | comment | added | quim | Fourth, seen this way, the rank 2 valuation defined above is a particular case of Zariski-Samuel, "Commutative Algebra II", VI.15, example 2, second paragraph. | |
| Sep 20, 2011 at 14:12 | comment | added | quim | Ouch, I now realize I should have said from the beginning that $C$ must be unibranch at $p$ (ie, analitically irreducible; this is always the case if $C$ is irreducible). | |
| Sep 20, 2011 at 14:11 | comment | added | quim | But, third! $C$ can be parameterized locally (analytically) by Puiseux series. That is, if $y,z\in {\mathcal O}_{S,p}$ are local coordinates at $p$ (equivalently, $y,z$ generate the maximal ideal at $p$) then there exist an integer $n$ and a series $h(t)$ such that $t\mapsto (t^n, h(t))$ is a parameterization of $C$ (i.e., the kernel of the map ${\mathcal O}_{S,p}\rightarrow {\mathbb C}[[t]]$ given by $y\mapsto t^n$, $z\mapsto h(t)$ is the ideal generated by $x$. Then, $I_p(C,\tilde f)$ can also be defined as the order (in t) of the image of $\tilde f$ in ${\mathbb C}[[t]]$. | |
| Sep 20, 2011 at 14:03 | comment | added | quim | Second, the intersection index $I_p(C,\tilde f)$ can be defined in various ways. One often found is as $\dim {\mathcal O}_{S,p}/(f,x)$. | |
| Sep 20, 2011 at 14:01 | comment | added | quim | I'll put several remarks in separate comments. First, the definition of that valuation: assume C is defined near p by x=0 for some $x\in {\mathcal O}_{S,p}$. Every $f\in {\mathcal O}_{S,p}$ can be written uniquely as $x^v⋅\tilde f$ with $v$ a nonnegative integer and $\tilde f$ not divisible by $x$ ($v$ is the order of $f$ over $C$). Now the map $f\mapsto (v,I_p(C,\tilde f))$ is a rank 2 valuation. | |
| Sep 20, 2011 at 13:26 | comment | added | Xin Nie | @quim: Thanks. As I could not find this example in Zariski-Samuel, can you give some details? I don't understand the meaning of counting "intersection with C at p", and "limit of all points on C infinitely near to p". | |
| Sep 20, 2011 at 9:23 | comment | added | quim | Easiest example: a complex curve. Every valuation is discrete of rank one, the Zariski-Riemann space is homeomorphic to the smooth model of C. This is the motivating example (no inverse limit needed, and well known before Zariski). First "interesting" case: surfaces. As an example, if p is a point of the curve C on S, a rank 2 valuation of K(S) counts order on C and intersection with C at p. The corresponding point in the Zariski-Riemann space is the limit of all points on C infinitely near to p. References: Zariski-Samuel "Commutative Algebra", Casas-Alvero "Singularities of plane curves". | |
| Sep 20, 2011 at 8:39 | history | asked | Xin Nie | CC BY-SA 3.0 |