19 votes

Rational curves on varieties of general type

I am surprised nobody mentioned the result of Lu and Miyaoka (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general ...
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15 votes
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When do 27 lines lie on a cubic surface?

It turns out that condition (T) is, indeed, sufficient for the $27$ lines (distinct and intersecting as expected) to lie on a cubic surface. To see this, consider the lines $a_1,a_2,a_3,a_4,a_5$ and $...
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  • 22.1k
15 votes

Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces

Weierstrass equations are probably a good choice. You can try $$y^2 = x^3 - 3x +2 t^{12},$$ for example. Here the singular fibers are when $t$ is a $24$th root of unity $\zeta$, and the double point ...
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14 votes

Restriction of the Picard group of a surface to a curve

Edit. I edited the argument below to make it work in all characteristics. By SGA $7_{II}$ Exposé XVII, this requires working with a sufficiently general pencil of divisors in $\mathcal{O}_{\...
14 votes
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Homeomorphism between del Pezzo surfaces

Yes, with precisely one exception. If $K^2 \neq 8$, then the del Pezzo surface is the blow-up of the plane at $9-K^2$ points, so it is homeomorphic to the connected sum of $\mathbb{CP}^2$ with $9-K^2$...
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14 votes
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Classification of smooth algebraic surfaces with a smooth morphism to $\Bbb P^1$

Let $k$ be an algebraically closed field. Let $f:X\to \mathbb{P}^1$ be a smooth proper morphism with fibres of dimension one. Note that the fibres of $f$ are geometrically connected by Stein ...
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12 votes
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Enriques surfaces over $\mathbb Z$

A preprint by Stefan Schröer came out today with the answer to this question: arXiv:2004.07025. No such Enriques surface exists. In fact, there is no classical Enriques surface over $\mathbb F_2$ ...
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12 votes
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Analogies between classical geometry on complex surfaces and Arakelov geometry

These are indeed good questions, and while there is a very good corpus of answers to them, the analogy is not perfect. 0. The non-archimedean analogy First of all, I would like to go back to the ...
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11 votes
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Where to find "Families of curves on a surface of general type" (MR0457450)?

My local library has the paper version, here is a scan.
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10 votes
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Automorphisms of del Pezzo surfaces

Not exactly. The quadratic transformation commutes with the action of $S_3$, and they both act on $(\mathbb{C}^*)^2$; so the automorphism group is $(\mathbb{C}^{*})^{2}\rtimes (S_3\times S_2)$. You'll ...
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10 votes
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Is it normal surface of general type to have infinitely many positive rank elliptic curves?

I am only posting this as an answer because it annoys me to see a question like this listed as "unanswered", thus "hovering" near the top of the list of unanswered questions. If dhy wants to write up ...
10 votes
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Algebraic surfaces with no deformations

There are several different notions of "rigidity" (local rigidity, global rigidity, infinitesimal rigidity, étale rigidity and strong rigidity) and it is possible to provide examples for each of them. ...
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10 votes
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A diffeomorphism of complex surfaces mapping subvarieties to subvarieties

I am posting this answer because the following linear algebra proposition is too long for a comment. Lemma. Let $A$, respectively $B$, be an invertible $\mathbb{R}$-linear operator on $\mathbb{C}^2$ ...
9 votes

Is it normal surface of general type to have infinitely many positive rank elliptic curves?

This is more or less what Jason has done, but maybe a bit more direct, and it is so elementary that it's hard not to call it an exercise that possibly does not belong on MO. Start with $y^2=x^4+z^6$. ...
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9 votes
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Additivity of Kodaira dimension for a nice fibration

There is another inequality which says the following: Easy addition (Using the same notation): $$ \kappa(X)\leqslant \kappa(X_y) + \dim Y $$ Consequently, if $Y$ is of general type, i.e., $\...
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9 votes
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Pull-back of an irreducible ample divisor via an isogeny of abelian varieties

I think the Proposition is not true if $D$ is singular. Take a smooth curve $C$ of genus 2, and $X=JC$; embed $C$ in $X$ (say, by choosing a point of $C$). Let $\alpha$ be a point of order 2 in $X$; ...
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9 votes

Intersection graph of $(-1)$-class divisors on surface of general type

On any surface $X$ with non-negative Kodaira dimension the $(-1)$-curves (i.e, the smooth rational curves $D$ with $D^2=-1$) are isolated, in other words any two of them do not intersect. From this ...
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9 votes
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Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

The diagonal $\Delta $ is linearly equivalent to $\{p\}\times \mathbb{P}^1 +\mathbb{P}^1\times \{p\} $ for any $p$ in $\mathbb{P}^1$. Therefore $X$ is the zero locus in $S\times S'$ of a section of $L:...
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9 votes

Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?

If $2$ is a cube mod $p$ then you can take $(x,y) = (ct^2, t^6-1)$ where $c^3 = -4$. This works for every odd $p \equiv -1 \bmod 3$, but since you specified $p \equiv +1 \bmod 3$ the first case is $p=...
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9 votes

Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?

You can observe that the elliptic curve $E_2: y^2=x^3+(t^3+1)^2$ is a generic fibre of a rational elliptic surface. Over the algebraically closed field $k$ the group of $k(t)$ points on the curve has ...
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9 votes
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Motivation for birational geometry

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 ...
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9 votes

Motivation for birational geometry

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: ...
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8 votes
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Is the Tate conjecture known for etale covers of products of curves

The answer to the last question (and therefore to the others) is yes. An étale cover of a variety $X$ is dominated by an étale Galois $G$-cover, for some finite group $G$; and this is given by a ...
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  • 34.4k
8 votes

Del Pezzo surfaces and homotopy groups of spheres

Borrowing from 4 different comments to make a complete answer (and making it community wiki): We can construct a bijection between the set of del Pezzo surfaces and $\pi_4(S_3)$ using topology. ...
8 votes
Accepted

Euler Characteristic of Real Algebraic Surfaces

A “formula” is a lot to ask for, but there are algorithms based on Morse theory. E.g. §5 of Basu (1999), or §3 of Fortuna-Gianni-Luminati (2004).
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8 votes
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Polars of algebraic curves and surfaces

Geometrically, one first meets polar hypersurfaces when studying tangent lines from a point to a hypersurface. In fact, this concept generalizes the classical polarity of a point with respect to a ...
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8 votes
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Are any of these complex surfaces ever projective?

Here is a simple method for constructing projective examples: Assume there exist maps $f:C \to \mathbb{P}^1$ and $g:T \to \mathbb{P}^1$ of the same degree which are totally ramified at $c$ and $t$. ...
7 votes
Accepted

A proper smooth surface is projective

Quoting from a very nice paper by Stefan Schroeer we have: "The criterion of Zariski [3, Cor. 4, p. 328] tells us that a normal surface $Z$ is projective if and only if the set of points $z \in Z$ ...
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7 votes
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Castelnuovo's rationality criterion on singular surfaces?

It does not hold in general: a cone over a smooth plane cubic satisfies $q=P_2=0$ but is not rational. On the other hand if $S$ has canonical singularities and $\tilde{S} \rightarrow S$ is any ...
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  • 34.4k
7 votes
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What's $H^*(X - \{x_1,\ldots,x_n\},\mathcal{O})$, when $X$ is a projective smooth surface?

The general reference for what follows is SGA 2, §1 to 4. Let $S\subset X$ be a finite subset, and $U:=X\smallsetminus S$. There is an exact sequence $$H^1_S(X,\mathcal{O}_X)\rightarrow H^1(X,\...
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