# Tagged Questions

**2**

votes

**1**answer

201 views

### Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...

**7**

votes

**1**answer

202 views

### Finite morphisms to projective space

Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...

**1**

vote

**1**answer

165 views

### Complete Linear system on Del Pezzo surfaces

Is there always a reducible curve (EDIT: with exactly two irreducible components intersecting in at least 2 points) in a complete linear system (EDIT: of dimension at least 2 with curves of genus at ...

**6**

votes

**0**answers

489 views

### Bezout Theorem in $\mathbb P^3$

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq ...

**10**

votes

**1**answer

623 views

### what is the cyclic cover trick?

What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explaination, both talking about curves and surfaces...

**1**

vote

**0**answers

126 views

### What does Hodge theory tell us about simply connected surfaces of general type

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...

**1**

vote

**1**answer

171 views

### Constructing a curve with good reduction over a function field

Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$.
Let $g\geq 0$ be an integer.
Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a ...

**2**

votes

**1**answer

354 views

### Genus two pencil in K3 surface

It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...

**0**

votes

**0**answers

166 views

### Does there exist a fibration of genus two over P^1 with only 3 singular fibres but two are semi-stable fibers for algebraic surfaces?

We will work over the complex numbers C.
there exist a fibration of genus two over P^1 with only 3 singular fibres but one is semi-stable fiber.
there not exist a fibration of genus two over P^1 ...

**5**

votes

**3**answers

588 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 ...

**2**

votes

**0**answers

227 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 ...

**10**

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 ...

**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 ...