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Birational geometry is extremely difficult for me to comprehend, because this type of transformation is so general and seems to preserve so little of the geometry. For example cubic surfaces are birational to hyperboloids, but the former class has 27 lines on it and the latter has infinitely many. I have little to no intuition regarding how birational maps behave geometrically and how one should "imagine" them, and would like to lessen that gap.

Now, for algebraic curves the situation isn't so bad, because we have the theorem that two smooth birational algebraic curves are isomorphic. I'm looking for similar results regrading algebraic surfaces - results that give insight (at least in special cases) about how close birational surfaces are to being isomorphic. I find the above theorem about curves to be a nice result because isomorphism of algebraic varieties does preserve geometric properties reasonably well - like singularities, local structure (Puiseaux expansion) etc, so I just think about smooth birational curves as having the same shape and it seems enough. For surfaces however, there is a very rich theory concerned with birationally classifying them, but I can't seem to understand what is the importance of this theory, because I can't imagine what is "similar" about birational surfaces. Are there similar (probably partial) results concerning the relationship between birational equivalence and isomorphism of algebraic surfaces? (I'm aware that every birational map is a sequence of blow-ups and blow-downs, but I don't really understand from that how similar birational surfaces are.)

roymend
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    I think part of your confusion with the $27$-lines vs $\infty$-many lines is that those are lines coming from an ambient projective space. So the number $27$ is not an invariant of cubic surfaces, but of cubic surfaces embedded into $\mathbb P^3$. It's a bit like saying "every $\mathbb P^1$ contains exactly $1$ line", but if you consider a conic $C \subset \mathbb P^2$, then this is abstractly isomorphic to $\mathbb P^1$, but does not contain any line coming from $\mathbb P^2$. – red_trumpet Jul 01 '21 at 11:58
  • @red_trumpet sure you need the embedding into P3 to state and prove the theorem about the 27 lines on cubic (because you need the embedding to even define what a cubic is). However, aren't the lines themselves intrinsic? In other words, isn't it true that any projective variety abstractly isomorphic to some cubic surface (which is embedded) also has 27 lines? – roymend Jul 01 '21 at 12:21
  • What I mean is that the notion of "being a line" for some curve $C \subset S$ is not an intrinsic property of the pair $(S, C)$. For example you could try to follow what happens to all the lines on a hyperboloid when applying the birational transformations to get a cubic surface in $\mathbb P^3$. I would guess they don't look like lines anymore (even though I expect them to still be isomorphic to $\mathbb P^1$, maybe some picking up singularities?) – red_trumpet Jul 01 '21 at 12:30
  • @red_trumpet but this example you gave (what happens to the lines on hyperboloid on birational transformation to cubic) doesn't show that the lines are not intrinsic, because the transformation you state is only birational, and not an isomorphism. This is exactly why I don't think the hyperboloid and cubic surface are abstractly isomorphic (even though they are birational). I don't have enough knowledge to attempt to prove this, though. – roymend Jul 01 '21 at 12:38
  • Actually Wikipedia states that cubics are isomorphic to the projective plane blown up in 6 points, so indeed they are not isomorphic to hyperboloids (as these are isomorphic to the projective plane itself, without blowups). – roymend Jul 01 '21 at 12:48
  • Well one way of thinking about birational geometry is not as "geometry" per se, but instead just asking questions about transcendental field theory. That is, you're asking for which varieties $X$, $Y$ we have $k(X) \cong k(Y)$. For collections of points, this is asking what the splitting field is. For curves, it's asking more complicated questions which end up boiling down to the isomorphism class of the normalisation. For surfaces everything is kinda crazy, but for example $k[x,y,z]/(cubic) \cong k[x,y]$. I'd say that in a way, you can think about birational transformations as being – Mummy the turkey Jul 01 '21 at 13:04
  • First note that a hyperboloid is $\mathbb P^1 \times \mathbb P^1$, which is actually the blow-up of $\mathbb P^2$ in one point. So we get (something isomorphic to) a cubic surface by blowing up a hyperboloid further. Then the strict transforms of the lines $\mathbb P^1 \times {q}$ will give infinitely many curves on the cubic surface, each of which is isomorphic to $\mathbb P^1$. And actually most of them don't even see the blow-up, because most of them don't contain the points that are blown-up. – red_trumpet Jul 01 '21 at 13:10
  • contd. "niceish especially on codimension $1$". For example, you're thinking about lines... if you instead think about genus $0$ curves then you'll find that (apart from possibly finitely many) a birational map plays prefectly nicely with those (similarly genus $g$ curves). – Mummy the turkey Jul 01 '21 at 13:11

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The structure theory is more complicated than for curves, but is still relatively concrete:

  1. Birational maps of surfaces factor as sequences of blow-ups at points.

  2. Every birational equivalence class of surfaces, other than the ruled surfaces (those birational to $C \times \mathbb{P}^1$ for $C$ a curve) contains a unique minimal model, which is to say a smooth surface $S$ that cannot be "blown down": any rational map $S \to S'$ is an isomorphism. (Ruled surfaces also have minimal models, but they are not unique.)

hunter
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