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Questions tagged [algebraic-surfaces]

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.

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39 votes
5 answers
3k views

Surfaces in $\mathbb{P}^3$ with isolated singularities

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...
Francesco Polizzi's user avatar
3 votes
1 answer
793 views

Variation of the Albanese map

Let $S$ be an irregular surface of general type over $\mathbb{C}$ and $a \colon S \to A:=\textrm{Alb}(S)$ be its Albanese map. Let Def($S$) and Def($A$) be the bases of the Kuranishi family of $S$ and ...
Francesco Polizzi's user avatar
15 votes
3 answers
2k views

A nontrivial surface on which any two curves intersect

One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, ...
Charles Staats's user avatar
5 votes
2 answers
413 views

Shrinking Fano surfaces to a point in Calabi-Yau 3-folds

Let X be a Calabi-Yau 3-fold, and $D\subset X$, smooth,Fano divisor. Since $K_X=0$ we have $N_D^X=K_D$. I have seen the following fact in many papers: By deforming X within Kahler moduli, we can ...
Mohammad Farajzadeh-Tehrani's user avatar
20 votes
3 answers
3k views

Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
user5395's user avatar
  • 545
7 votes
1 answer
799 views

Liftability of Enriques Surfaces (from char. p to zero)

Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$. We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
Holger Partsch's user avatar
8 votes
3 answers
3k views

Cone of curves and Mori theorem for algebraic surfaces

In describing part of the geometry of the cone of curves for an algebraic surface $S$, we need to find $(-1)$ curves within $S$. Once we've done that, then we can say that the "negative" ...
Csar Lozano Huerta's user avatar
27 votes
5 answers
7k views

blowing up, -1 curves, effective and ample divisors

Lets say we're on a smooth surface, and we blow up at a point. Is there a simple explicit computation that shows to me the fact that the exceptional divisor E has self intersection -1 ? I don't ...
fellow's user avatar
  • 271
5 votes
1 answer
6k views

Algebraic equivalence VS Numerical Equivalence - An Example.

This question is arose from the question Difference between equivalence relations on algebraic cycles and the example 3 in lecture 1 in Mumford's book Lectures on curves on an algebraic surface. ...
Fei YE's user avatar
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6 votes
4 answers
873 views

Interaction of topology and the Picard group of Algebraic surfaces

It is well known that a smooth cubic surface $X\subset \mathbb{P}^3$ has exactly 27 lines in it. Furthermore, it is easy to check that Picard group $$Pic(X)\cong \mathbb{Z}^7$$ Here the generators are ...
Csar Lozano Huerta's user avatar
10 votes
1 answer
2k views

When is the canonical divisor of an algebraic surface smooth?

Is there some condition on a complex algebraic surface that implies it has a smooth canonical divisor? I am searching for the sharpest possible condition, but sufficient criteria would be nice as ...
James O's user avatar
  • 445
4 votes
1 answer
988 views

Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces

I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$ (such thing is called an ACM surface, I think) and a globally ...
Hailong Dao's user avatar
  • 30.5k
16 votes
4 answers
1k views

K3 surfaces with good reduction away from finitely many places

Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...
JSE's user avatar
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12 votes
3 answers
2k views

level sets of multivariate polynomials

Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
ioannis.parissis's user avatar
11 votes
3 answers
2k views

Nonprojective Surface

Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and ...
Charles Siegel's user avatar
15 votes
6 answers
3k views

Curves with negative self intersection in the product of two curves

I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative ...
Dmitri Panov's user avatar
  • 28.9k
3 votes
1 answer
514 views

K3 over fields other than C?

How to classify K3 surfaces over an arbitrary field k?
Ilya Nikokoshev's user avatar
4 votes
1 answer
1k views

The existence of primitive and sufficiently ample line bundles on K3 surfaces?

Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is ...
user761's user avatar
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