19
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 $...
- 32.5k
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 ...
- 148k
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 ...
- 9,497
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 ...
- 12.9k
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.
- 6,871
11
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:...
- 38k
11
votes
Accepted
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
The question is essentially asking why the inclusion of $Z$ into the smooth locus $Y$ induces a surjection on $\pi_1$. The ambient space $X$ is not really relevant here. You're removing some Zariski ...
- 8,803
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., $\...
- 42.9k
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.
...
- 66.3k
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 $\mathbb{C}^2$ ...
10
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 ...
- 148k
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 ...
- 9,580
10
votes
What is the smallest and "best" 27 lines configuration? And what is its symmetry group?
It is the Fermat cubic surface over $\mathbb{F}_4$ or (if you prefer $\mathbb{F}_p$) over $\mathbb{F}_7$.
There are quite a few papers on this topic. Firstly in [1] Swinnerton-Dyer showed (amongst ...
- 21.3k
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 $X$; ...
- 38k
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 ...
- 66.3k
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=...
- 79.9k
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 ...
- 303
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 $t$. ...
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 ...
- 42.9k
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. ...
- 1,306
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, ...
- 375
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 ...
- 66.3k
8
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 $\...
- 1,144
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 ...
- 28.7k
7
votes
Accepted
Infinitely many exceptional curves on ruled surfaces
Let me write a short answer summarizing the comments above, so that the question will not appear unanswered anymore.
Proposition. The following holds.
(1) If $C$, $D$ are smooth curves and $g(C) \...
- 66.3k
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 ...
- 14.3k
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...).
...
- 28.7k
7
votes
Accepted
Cohomology of singular projective cubic surface
By the classification theorem of cubic surfaces (p.6 in this paper), a cubic surface belongs to the following classes
Has at worst ADE singularities.
Has an elliptic singularity, i.e., the surface ...
- 1,803
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 (...
- 3,409
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