The algebraic-surfaces tag has no usage guidance.

**2**

votes

**2**answers

695 views

### Isolated conics on a del Pezzo surface

Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated ...

**3**

votes

**1**answer

429 views

### Mordell-Weil Group of Elliptic Surface

Suppose $ X \to \mathbb{P}^1 $ is an elliptic surface with section, with Weierstrass model defined over $ \mathbb{Q} $. If $ \sigma: \mathbb{P}^1 \to X $ is a torsion section with order $n$, then for ...

**1**

vote

**0**answers

204 views

### On base locus of canoncal linear system on surfaces

Let $S$ be a minimal surface of general type over $\mathbb{C}$ with $p_g=h^0(K_S)>1$. As a convention, we can write $|K_S|=|M|+F$ such that $F$ is the fixed part. We know that $K_SM \le K_S^2$.
...

**6**

votes

**2**answers

595 views

### Embedding of algebraic surfaces

If I am not mistaken there is a theorem that says any curve $C$ can be embedded in $\mathbf{P}^3$. What can be said about surfaces? Do we have a theorem like:
All surfaces can be embedded in ...

**7**

votes

**2**answers

1k views

### rational curve on varieties of general type

Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?

**5**

votes

**3**answers

646 views

### Are there (-2)-curves on an Enriques surface?

Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it ...

**8**

votes

**1**answer

390 views

### Are there non-projective normal surfaces which are rational?

Every non-singular complete surface is projective. On the other hand, there are non-projective complete surfaces (see e.g. Excercise II.7.13 of Hartshorne) - and there are such examples where the ...

**21**

votes

**0**answers

543 views

### bound on the genus of a fiber of the Albanese map of a surface with $h^1({\mathcal O})=1$?

This is maybe more an open problem than a question, since I have seriously thought about it and asked several people working on algebraic surfaces with no success. I hope somebody here can ...

**13**

votes

**4**answers

1k views

### Algebraic surfaces and their (intrinsic) geometry

Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...

**2**

votes

**1**answer

321 views

### Quotient of an abelian surface by a finite group, irreducible components

Given an abelian surface $A$ as the product of two non isogenous elliptic curves $E_1 $ and $E_2$ over $\mathbb{C}$. We also have a smooth 2 dimensional moduli space $M$ of sheaves on $A$ with some ...

**5**

votes

**3**answers

587 views

### Basepoints in the Canonical System of Algebraic Surfaces

Let X be an smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...

**4**

votes

**1**answer

333 views

### Certain double covers of cubic surfaces

Let $S$ be a smooth cubic surface in $\mathbb{P}^3$. I would like to understand that variety $V$ that parametrizes lines $\ell$ such that $\ell \cdot S=3P$ with $P \in S$. At any point $P \in S$, let ...

**5**

votes

**1**answer

642 views

### an exercise about elliptic surface in Beauville's book

In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian ...

**20**

votes

**4**answers

2k views

### Abundance for algebraic surfaces

I am currently teaching a course in algebraic geometry where one of the aims is to give an overview of the Enriques-Kodaira classification of surfaces. I am trying to throw in some modern aspects so I ...

**5**

votes

**2**answers

636 views

### Ramification in morphisms of surfaces

The question can be generalized, but we might as well restrict to this case.
Let $X \rightarrow Y$ be a morphism between nonsingular surfaces (say over $\mathbb{C}$). Let $R_1$ be an irreducible ...

**2**

votes

**0**answers

236 views

### quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...

**7**

votes

**1**answer

391 views

### How do branched coverings of complex surfaces “fit” with branched coverings of curves?

Since I'm used to working with algebraic $\pi_1$'s, which don't work well with surfaces, I find myself lacking geometric intuition when I attempt to do these types of purely geometric arguments. I'm ...

**31**

votes

**5**answers

2k 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, ...

**3**

votes

**1**answer

623 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 ...

**12**

votes

**3**answers

1k 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 $W \geq 0$, ...

**4**

votes

**2**answers

325 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 ...

**11**

votes

**3**answers

2k 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 ...

**7**

votes

**1**answer

306 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 ...

**5**

votes

**3**answers

2k 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" part of the ...

**13**

votes

**5**answers

3k 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 ...

**2**

votes

**1**answer

2k 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.
...

**6**

votes

**4**answers

554 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 ...

**10**

votes

**1**answer

952 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 ...

**4**

votes

**1**answer

474 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 ...

**13**

votes

**4**answers

994 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 ...

**9**

votes

**3**answers

910 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 ...

**7**

votes

**3**answers

974 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 ...

**13**

votes

**6**answers

2k 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 ...

**3**

votes

**1**answer

441 views

**3**

votes

**1**answer

585 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 ...