Timeline for Motivation for birational geometry
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2021 at 15:02 | vote | accept | roymend | ||
| Jul 21, 2021 at 7:17 | answer | added | Sándor Kovács | timeline score: 9 | |
| Jul 21, 2021 at 1:49 | answer | added | Will Sawin | timeline score: 10 | |
| Jul 20, 2021 at 14:17 | comment | added | roymend | @SándorKovács OK, fair enough. Let me pose a follow-up question that I think will help me understand better what makes birational geometry interesting\worthwhile (and is really a refinement of the original question): what are some examples of problems studied in birational geometry, except the problem of birational classification of varieties? I mean, given that so much work went on birational classification of varieties, what are some interesting properties of varieties that are preserved under birational transforms? Rational curves on varieties were already mentioned as one good example. | |
| Jul 20, 2021 at 0:35 | comment | added | Sándor Kovács | OK, let me try one more time: I think you are approaching this from the wrong angle. As @dhy already said it, any higher dimensional object is not suitable for visualization as that is understood traditionally. On the other hand, functions are part of geometry. How do you distinguish between topology, differential geometry, complex geometry, or algebraic geometry? The difference is the choice of preferred functions. Functions determine the kind of geometry you are doing. They also help us represent geometric objects that are hard to visualize because our physical reality is limited. | |
| Jul 19, 2021 at 21:22 | history | edited | YCor | CC BY-SA 4.0 |
removed "soft question" which doesn't really fit its usual meaning here, changed to big-picture
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| Jul 19, 2021 at 21:19 | comment | added | roymend | @dhy I actually agree, rational curves on surfaces are a very good example of a geometric question for which the tools of birational geometry are relevant. Thanks, and I'm eager for more such examples if there are any. (I'm aware that birational geometry is in general quite relevant for some questions in enumerative geometry, so I'd prefer non-enumerative examples.) | |
| Jul 19, 2021 at 21:10 | comment | added | dhy | ...(reality is not quite as clean as this outline, but hopefully it shows how useful birational geometry is.) I think all these considerations are geometric in nature. At the very least, it's very hard for me to see any of the above reasoning as algebraic (no equations show up anywhere). Maybe it doesn't fit "easily visualized" (because algebraic surfaces have 4 real dimensions) but in that case I'm not sure anything in algebraic geometry past curves is geometric by your definition. | |
| Jul 19, 2021 at 21:04 | comment | added | dhy | ...very important for understanding the behavior of such curves. E.g., if I want to classify rational curves on some given surface $X$, the first step to do so is to know where in the classification of surfaces $X$ fits into. Say for instance that $X$ is rational.... then the classification tells me that it's an iterated blowup of $\mathbb{P}^2$ or a Hirzebruch surface, so if I can understand 1. curves on those surfaces and 2. how the behavior changes upon blowing-up then I can say a lot about rational curves on any rational surface... | |
| Jul 19, 2021 at 20:52 | comment | added | dhy | Re: "given a picture of a surface without its equation, I doubt that you'll be able to tell whether it is rational, Enriques or K3." Sure I can: if the surface contains families of rational curves, it's rational. (OK, this is a bit facetious, since maybe you give me a picture where I can't tell if it contains rational curves.) It's still unclear to me what you consider geometric, but since you mention enumerative geometry, maybe you'd include the study of rational curves on varieties? In which case it is worth mentioning that birational geometry is... | |
| Jul 19, 2021 at 20:51 | comment | added | Sándor Kovács | @roymend: functions are geometric. A stereographic projection is a function with a certain property. The beauty and intrigue of algebraic geometry is that it unifies algebra and geometry. You can view the same question algebraically or geometrically and get a much better understanding of both sides. | |
| Jul 19, 2021 at 19:45 | history | edited | roymend | CC BY-SA 4.0 |
edited body
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| Jul 19, 2021 at 19:42 | comment | added | roymend | That a conic can be stereographically projected to a line. Stereographic projection can be easily visualized so this gives nice geometric information. An example that I don't consider geometric is the birational classification of surfaces: given a picture of a surface without its equation, I doubt that you'll be able to tell whether it is rational, Enriques or K3. @SamHopkins, thanks a lot! I'll certainly read it. | |
| Jul 19, 2021 at 19:39 | comment | added | roymend | ...The theorem that two triangles with the same angles are similar (again you can visualize the transformation). That two plane curves with the same curvature function are congruent. I've limited myself to theorems about transformations to try to be relevant to birational geometry; of course there are many other theorems I consider geometric that are not about transformations (say Pascal's theorem, or Bezout's, or anything in enumerative geometry). Let me also mention the only birational theorem I know of that I do consider geometric: | |
| Jul 19, 2021 at 19:39 | comment | added | roymend | @dhy I'm looking for theorems (or questions) that can be easily visualized. That are "about shapes" (except for topological stuff). I'll give some examples, but this is inherently misleading, since every example I know will necessarily talk about properties of shapes that are not birational, but instead come from another flavor of geometry (projective, euclidean, or even differential). So for example, the theorem that any two conics are projectively equivalent to each other (you can really visualize moving the line at infinity to a finite place, transforming a parabola into an ellipse)... | |
| Jul 19, 2021 at 18:16 | comment | added | dhy | Can you give examples of questions you consider "geometric"? (I suspect the answer to your question will overwhelmingly be that birational geometers think of the field geometrically, but your question seems to imply that you don't consider something like "are these two varieties birationally equivalent?" a geometric question.) | |
| Jul 19, 2021 at 17:36 | comment | added | Sam Hopkins | I think the article "Algebraic hypersurfaces" by J. Kollár (ams.org/journals/bull/2019-56-04/S0273-0979-2019-01663-2/…) does a really good job motivating the study of birational isomorphism, with a particular focus on the case of hypersurfaces. | |
| Jul 19, 2021 at 17:27 | review | First posts | |||
| Jul 19, 2021 at 17:31 | |||||
| Jul 19, 2021 at 17:19 | history | asked | roymend | CC BY-SA 4.0 |