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](https://math.stackexchange.com/questions/4202459/soft-question-motivation-for-birational-geometry), but was advised to move it here.



Motivation for birational geometry

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




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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.


