20
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 ...

16
votes

Accepted

### 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 $...

15
votes

Accepted

### 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 $\mathbb F_2$ ...

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 ...

14
votes

Accepted

### 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 $9-K^2$...

14
votes

Accepted

### 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 ...

13
votes

Accepted

### 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 ...

11
votes

Accepted

### Where to find "Families of curves on a surface of general type" (MR0457450)?

My local library has the paper version, here is a scan.

10
votes

Accepted

### 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., $\...

10
votes

Accepted

### 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.
...

10
votes

Accepted

### 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 $\mathbb{C}^2$ ...

Community wiki

10
votes

Accepted

### Topology change induced by small perturbation

The formal statement you are thinking of when you assert "The topology changes only when..." is Ehresmann's theorem: a proper smooth submersion is a fiber bundle, and hence all fibers are ...

9
votes

Accepted

### 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 $X$; ...

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 ...

9
votes

Accepted

### 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:...

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=...

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 ...

9
votes

Accepted

### 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 $t$. ...

Community wiki

9
votes

Accepted

### 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 ...

9
votes

### Motivation for birational geometry

Will has already said many of the things that 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 ...

9
votes

Accepted

### What is the surface area of the finite part of the Cayley nodal cubic surface?

Here is the start of an answer - I am afraid that I don't want to spend any more of my time thinking about this problem, but it gets the answer in terms of an integral that looks very computable. ...

8
votes

Accepted

### General conditions for normality of blow-up

Let $X=Spec(R)$. Blowing-up $Z=V(I)$ is the same as to look at $Proj$ of the graded ring $R[It]=\oplus_{j\geqslant 0} I^jt^j\subset R[t]$, the Rees ring associated to $I$.
Assume $R$ is a domain, ...

8
votes

Accepted

### Volume of a divisor on a smooth projective surface

At least for effective divisors, the answer is strongly related to Zariski decomposition.
If $D$ is an effective divisor on a smooth surface $X$, Zariski proved in [Z62] that there exists a unique ...

8
votes

### A constructive proof of the theorem of the cube

This is not really an answer, but a rephrasing together with some comments on why this is difficult. I end with one example where you can actually compute something (purely algebraically) on $E \times ...

7
votes

Accepted

### 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,\...

7
votes

### Smoothness of the branch divisor and ramification on surfaces

It seems to me that the intersection is zero in general, i.e. the answer is YES.
Let us prove that $f^{-1}(B)$ is a smooth curve in $X$. This clearly implies the desired result. The proof of ...

7
votes

Accepted

### Intuition behind results in Mumford's "Lectures on curves on an algebraic surface", I

I think I can provide some intuition for (A), both in characteristic $0$ and $p > 0$. What follows below is more or less a proof, but with a lot of omissions (and hopefully not too many lies...).
...

7
votes

Accepted

### Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?

Please find below a short argument in the case of schemes. The answer is positive for algebraic spaces too; in that case it can be proven using the characterization: $X$ has the resolution property $\...

7
votes

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

As far as I know no one has “written down” a hyperkähler structure on a K3 surface. Indeed much of the work in Mirror symmetry revolves around trying to give an asymptotic expansion of such metrics (...

7
votes

Accepted

### Multiplicity of irreducible component of a singular fiber of a $\mathbb{P}^1$-fibration

If we assume that the rational curves intersect in nodes, then we can compute it from the data of the dual graph. Since $L$ intersects only one of the $D_i$, then the $D_i$ necessarily form a chain, ...

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