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I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that interest them?

Often I hear that the main objective of birational geometry is to classify algebraic varieties up to birational isomorphism. This is an interesting mathematical question, but do the people who study it view it as a geometric question or as a commutative-algebraic one?

As a concrete example, we can take the birational classification of surfaces: it gives you very nice answers to questions like "when can we find a function with certain properties between two surfaces? what are the properties of these functions?" However, I don't see what kinds of geometric questions (i.e. questions about shapes) this theory gives you answers for, given that surfaces birational to each other can nevertheless look so radically different from each other.

Do people who study birational geometry view their field as geometry or as algebra? What are the kinds of geometric questions that interest them?

I originally asked this question on math.se, but was advised to move it here.

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    $\begingroup$ 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. $\endgroup$ Jul 19, 2021 at 17:36
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    $\begingroup$ 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.) $\endgroup$
    – dhy
    Jul 19, 2021 at 18:16
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    $\begingroup$ @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. $\endgroup$ Jul 19, 2021 at 20:51
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    $\begingroup$ ...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... $\endgroup$
    – dhy
    Jul 19, 2021 at 21:04
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    $\begingroup$ 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. $\endgroup$ Jul 20, 2021 at 0:35

2 Answers 2

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what are some interesting properties of varieties that are preserved under birational transforms?

I will answer the question for smooth projective varieties (certainly a geometrically nice class of varieties) specifically.

(1) For each $k$, the dimension of the space of global holomorphic/algebraic differential $k$-forms is a birational invariant. This is one of the most important invariants that is very meaningful both algebraically and geometrically, and it itself plays a significant role in the classification of birational types of surfaces.

(2) The fundamental group $\pi_1(X)$ is a birational invariant. This is certainly one of the most fundamental topological invariants of the variety.

(3) For $Y$ any variety not containing rational curves, the set of maps $X \to Y$ (or the moduli space parameterizing maps) is a birational invariant.

Of course, maybe a more general answer is that finding interesting properties of varieties preserved under birational transforms is (by definition) one of the main areas of study of birational geometry! This is often done with the goal of, say, proving a particular variety is not rational, but you can also view these as more new invariants which you now know you understand after determining the birational type of a given variety.

Another more general answer is "often, almost everything". In particular, for many birational equivalence classes (almost all in the case of surfaces), we can identify inside the birational equivalence class a single variety $Y$ such that every smooth projective variety $X$ in the equivalence class must in fact map birationally to $Y$. We can, for many purposes, think of the variety $X$ as just $Y$ with a bit of extra elaboration added. Finding the most general statement of this type is a primary goal of the minimal model program.


To answer your original question, the birational classification of surfaces forms a starting point to answering the majority of questions about the existence of a surface with a particular property. If called upon to construct a surface with some unusual features, an experienced algebraic geometer will, if an obvious strategy for constructing one doesn't present itself, automatically start running through "Can a rational surface have this property? What about ruled? K3? General type? Elliptic? ...", considering blow-ups of these as appropriate.

Such a coarse classification is needed for getting the lay of the land, ruling out wide swathes of terrain to direct your attention to more fertile fields for growing surfaces of a particular type. If you had to start listing "What about the plane blown up at one point? What about the plane blown up at two points? What about the plane blown up at three points? ..." and so on, you'd never get anywhere, except for those particular problems where cleverly blowing up the plane is the linchpin to a brilliant solution.

I am not a birational geometer, but I gained a great appreciation for the birational classification of surfaces by playing around with different problems, on MO and elsewhere. I can't imagine organizing information on surfaces in a useful way without it.

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    $\begingroup$ There is a bit of overlap between this answer and Tabes Bridges answer yesterday on math.SE. $\endgroup$
    – Will Sawin
    Jul 21, 2021 at 1:50
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    $\begingroup$ For (2) , you probably meant to say that $X$ is nonsingular and probably projective (or at least proper), no? (A nodal cubic has a nontrivial fundamental group and so does $\mathbb C^\times$, both birational to $\mathbb P^1$). $\endgroup$ Jul 21, 2021 at 6:41
  • $\begingroup$ @WillSawin, thank you very much for the detailed answer, which is certainly helpful to me and offers some motivation. Do you have any suggestions for references (including books) where I can look for such interesting problems about surfaces (or other varieties) as you mentioned? $\endgroup$
    – roymend
    Jul 21, 2021 at 9:36
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    $\begingroup$ @SándorKovács Yeah, I was thinking of smooth projective varieties. I had just answered another question where the asker specified they were only interested in smooth projective varieties, and I somehow got it mixed up with this one. $\endgroup$
    – Will Sawin
    Jul 21, 2021 at 12:00
  • $\begingroup$ @roymend Thanks! Unfortunately, I can never come up with good references to things, and thus am not able... $\endgroup$
    – Will Sawin
    Jul 21, 2021 at 12:01
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Will has already said many of the things I would have said trying to answer your extended question, but let me add a few things without trying to avoid overlap.

In fact, let me start with an overlap: Indeed, a fundamental question of birational geometry is what you asked: "What are some interesting properties of varieties that are preserved under birational transforms?" Studying this question is already interesting, but let me say a bit more.

Algebra vs. Geometry I know this is the original question, but I would like to record my favorite comment about this issue. I would actually say that (at least with respect to algebraic geometry, and hence to birational geometry), Algebra=Geometry!

Let me give two quick examples:

a) Resolution of singularities of curves, a.k.a., starting with an arbitrary (=possibly singular, non-compact) curve and constructing a smooth projective curve which is birational to the original one. You might agree that this is a "purely" geometric question. However, one way to do it is to consider the set of DVRs of its function field (plus some technical condition), define first a topology, then regular and rational functions (functions, again!) on this set and then prove that this set is actually isomorphic to an "actual" curve.

b) Lüroth's theorem, which asserts that if $k$ is an algebraically closed field and $k\subsetneq K \subseteq k(x)$ is a subfield of the purely transcendental extension of $k$ of transcendence degree $1$ is isomorphic to $k(x)$ (i.e. $K\simeq k(x)$, but not necessarily equal). You might agree that this is a "purely" algebraic question. Yet, one way to prove this is to observe that $K$ corresponds to a smooth projective curve, say $X$, over $k$ (for instance by a)!) and the inclusion $K \subseteq k(x)$ induces a morphism $\mathbb P^1\to X$. The latter implies that $X\simeq \mathbb P^1$ and hence the statement of the theorem.

These two examples also provide examples to your extended question:

A few things where birational geometry is relevant/interesting/worthwhile:

  1. Resolution of singularities: Given an arbitrary variety $X$, find a proper/projective morphism $\tilde X\to X$ such that $\tilde X$ is non-singular. This has lots of variants, and the relevant existence theorems are probably the most often used ones in algebraic geometry.

  2. Rationality questions: Which algebraic varieties are birational to projective space? If you want, this is a generalization of Lüroth's theorem. This is also one that one can consider either a purely geometric or a purely algebraic question. Again, lots of variants. Interesting keywords to look up: unirational, rationally connected varieties.

  3. The minimal model program is a quintessential part of birational geometry. It is sometimes considered part of classification, although it is actually really a precursor of that. The main guiding question of the mmp is: How can we choose a nice representative from every birational class?

These are the first three that come to my mind, but others might argue that I am forgetting some. So, let me add some random examples without claiming that they are the most important ones.

  1. Birational geometry comes up in string theory. Physicists like to work with smooth objects, so they are in trouble when they encounter a singular object, which is essentially inevitable. So they want to make is smooth. You can do that either by resolution of singularities (as above) or (sometimes) by deforming it to something smooth. Doing both leads to something they call the conifold transition (see here)

  2. Another place where birational geometry is extremely important is moduli theory. This is another huge subject, so I'm not going to get into it.

In fact, I kind of feel that I could just keep going, so instead I will stop here. Let me just say this: birational geometry is everywhere in algebraic geometry and even beyond that.


To respond to the question in the comments:

I would somewhat disagree with your assessment of the message of the answer on math.se. IMHO, isomorphism and birational equivalence are not competing.

A different way to say what the math.se answer is saying is this: A priori the classification problem is to classify all varieties up to isomorphism. This is essentially impossible in the sense of usual classifications (such as classification of finite simple groups).

The method employed is the following: first find a nice representative in each birational class, and a way to obtain it, then classify these nice representatives (up to isomorphism). This does give you a sort of classification of all varieties up to isomorphism. Start with your favorite one, obtain the nice representative, look it up in the classification. Now, you can obtain the original one by reversing the way you obtained the nice representative starting with the one you found on the list.

In fact, this gives you a better classification than if you just had a list, because using this method you might be able to compare two varieties and decide whether they are isomorphic and if not then whether they are birational. But this is not the main point.

The main point is what I said already, isomorphism and birational equivalence are not in competition. They play for the same team.

(Also, just to link this to the above. That "nice representative" I mention here is the minimal model and the way to get it is provided by the mmp.)

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    $\begingroup$ In the answer to the version of the question on math.se, it is suggested that isomorphism is really what's significant, and birational equivalence is used as a step towards understanding isomorphism. Do you agree with that? Or would you say it's more like homeomorphism versus homotopy equivalence, where algebraic topologists have learned that the equivalence notion of real interest is not the naive one (perhaps because their techniques work best for the more sophisticated notion)? $\endgroup$ Jul 21, 2021 at 11:18
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    $\begingroup$ I second @SamHopkins's question, and would really like to hear opinions on this. By the way, Kollar's article Sam linked to in a comment above, also seems to suggest that isomorphism is really the more valuable notion (theorem 30 there, which I found to be the most useful in terms of helping me understand what is geometric here, essentially says that for hypersurfaces, isomorphism is almost the same as projective equivalence. But then there's also theorem 36, which says a similar thing for birationality.) $\endgroup$
    – roymend
    Jul 21, 2021 at 11:56
  • $\begingroup$ @SamHopkins: I started answering in the comments, but it started to be pretty long, so I will do it within the original answer... $\endgroup$ Jul 22, 2021 at 2:44
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    $\begingroup$ @roymend: 1) Who is that person "Kollar" you speak of? I never heard that name. 2) Theorem 30 in the linked article by Kollár is indeed about isomorphisms, but that says nothing about the importance of birational equivalence. $\endgroup$ Jul 22, 2021 at 2:58

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