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Results tagged with algebraic-surfaces
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user 5101
An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
13
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 oth …
- 21.4k
4
votes
Smooth projective surface with geometrically integral reduction
Let $X$ be an integral regular scheme which is proper over $\mathbb{Z}_p$. Assume that the special fibre $X_{\mathbb{F}_p}$ is irreducible and let $k$ be the algebraic closure of $\mathbb{F}_p$ in the …
- 21.4k
5
votes
A property of varieties between unirational and retract rational
Let $k$ be a field of characteristic $0$, $a \in k$ and $f$ a separable polynomial of degree $3$.
The projective surface $X$, given as the minimal smooth compactification of the affine surface
$$X: …
- 21.4k
3
votes
Field of definition for general type surfaces
Firstly, since you are interested in the field of definition of the surfaces, you should work with the moduli stack, rather than the coarse moduli space. A $k$-rational point on the moduli stack corre …
- 21.4k
6
votes
1
answer
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Del Pezzo surfaces of degree $2$
I'm trying to understand the relationship between the different models of del Pezzo surfaces of degree $2$.
Let $k$ be a field of characteristic not equal to $2$. Usually, del Pezzo surfaces of degre …
- 39.3k
3
votes
Accepted
Conjugate surfaces: informations about the orbits
This problem is of a more arithmetic nature, than geometric.
For example, since every automorphism of $\mathbb{C}$ preserves $\mathbb{Q}$, we see that if $X$ can be defined over $\mathbb{Q}$, then $\ …
- 21.4k
5
votes
Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$
Here is another way to see this. The automorphism group of any quadric hypersuface $$Q(x) = 0 \subset \mathbb{P}^n,$$ is exactly the projective orthogonal group $\textrm{PO}(Q)$ of $Q$.
The key poin …
- 21.4k
4
votes
0
answers
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views
An arithmetic analogue of the discriminant curve of a conic bundle threefold
I am looking for an "arithmetic" analogue of a well known result on threefolds with a conic bundle structure. The following result can be found in [Iskovskikh - On the rationality problem for conic bu …
- 21.4k
5
votes
Algebraic surfaces and their (intrinsic) geometry
Any non-singular complex variety $V$ of dimension $n$ (in either affine space or projective space) can be endowed with the structure of a complex manifold of dimension $n$. Moreover as a submanifold o …
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