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I'm interested in asking 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 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?

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    $\begingroup$ You can also post this in mathoverflow, (if you haven't yet) for more ideas, motivation and insight $\endgroup$
    – A S D
    Jul 19, 2021 at 17:02
  • $\begingroup$ Please give a (valuable) reference for people that are familiar with a certain number of geometries but who haven't the faintest idea of what birational geometry is... $\endgroup$
    – Jean Marie
    Jul 19, 2021 at 21:06
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    $\begingroup$ @JeanMarie the algebraic geometry tag should indicate that this question is aimed at algebraic geometers; birational geometry is a major subfield of AG, but it's difficult to explain (beyond just stating a few definitions) if one does not already know the basic lay of the land, standard toolkit, and routine bag of examples in AG. $\endgroup$ Jul 20, 2021 at 7:13

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The fundamental motivation is the classification of algebraic varieties up to isomorphism ("biregular classification"). Birational geometry gives us a natural way to break this hopelessly complex and difficult problem into two (slightly less hopeless) pieces: 1) classify varieties up to birational equivalence (i.e. classify field extensions of a given dimension over your ground field), and 2) understand what the possible birational models of a given function field look like. This latter question is addressed by things like the Minimal Model Program due to innumerable authors (Mori, Kollár, Shokurov, Birkar-Cascini-Hacon-McKernan,...) and the Weak Factorization Theorem of Abramovich-Karu-Matsuki-Wlodarczyk.

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