**29**

votes

**10**answers

2k views

### Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...

**17**

votes

**5**answers

4k 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 ...

**15**

votes

**2**answers

1k views

### Bertini theorems for base-point-free linear systems in positive characteristics

Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...

**14**

votes

**6**answers

2k views

### Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...

**14**

votes

**1**answer

329 views

### Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...

**13**

votes

**2**answers

3k views

### Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow ...

**12**

votes

**2**answers

3k views

### does a line bundle always have a degree

For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...

**12**

votes

**0**answers

791 views

### Relative canonical divisors

Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$.
In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if ...

**11**

votes

**2**answers

395 views

### Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...

**11**

votes

**0**answers

1k views

### Ample divisors on projective surfaces

Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$?
Background: I was reading ...

**10**

votes

**3**answers

2k views

### Contracting divisors to a point

This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...

**9**

votes

**1**answer

458 views

### Blowing-up a point in the singular locus

Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...

**9**

votes

**2**answers

314 views

### Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension).
Let $g:X\rightarrow ...

**9**

votes

**2**answers

218 views

### Effectiveness of the distinguished theta characteristic in characteristic 2

Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic ...

**8**

votes

**2**answers

939 views

### Anticanonical divisor of the blow up of P^2 in 9 points

Let $S$ the blow up of $P^2$ in nine points. Why is the anticanonical divisor $-K_S$ not semiample?

**8**

votes

**1**answer

451 views

### Does combining Abhyankar's Lemma and embedded resolution give horizontal normal crossings

Let $\pi:Y\longrightarrow \mathbf{P}^1_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch ...

**8**

votes

**2**answers

638 views

### Base locus of divisors on blowings up of the projective space

Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position.
Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with ...

**7**

votes

**2**answers

3k views

### The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...

**7**

votes

**2**answers

500 views

### How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function? [duplicate]

In order to know more about product over primes ,I would like to know how do I show that :$$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$$ without using properties of Riemann zeta function ?
Note01 : it ...

**7**

votes

**1**answer

556 views

### Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...

**7**

votes

**2**answers

515 views

### Irreducible divisors containing an arbitrary closed set

Let $X$ be a normal complex projective variety, let $V$ be a closed subset of $X$ (possibly reducible), and let $I_V$ be its ideal sheaf (consider the reduced scheme structure for example).
If $A$ is ...

**7**

votes

**1**answer

423 views

### Cones, monoids, and the space of (very) ample divisors

An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...

**7**

votes

**1**answer

830 views

### Why is (line bundle, appropriate rational section) not a standard kind of divisor?

In algebraic geometry, there are two standard "kinds" of divisors: Weil divisors and Cartier divisors. Weil divisors provide better geometric intuition, while Cartier divisors are more general (if ...

**6**

votes

**4**answers

896 views

### Cone over the Veronese surface

Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = ...

**6**

votes

**1**answer

188 views

### Self-intersection of a Cartier divisor

Let $X$ be a smooth projective variety, and $D$ a Cartier divisor on $X$ inducing a surjective morphism $f\colon X\rightarrow C$, where $C$ is a curve.
May we conclude that $D^{2}=0$?

**6**

votes

**3**answers

889 views

### Cone of movable curves

Let $X$ be a smooth complex projective variety of dimension $n$.
Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure ...

**6**

votes

**1**answer

505 views

### Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer?

After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for ...

**5**

votes

**3**answers

701 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 ...

**5**

votes

**3**answers

423 views

### Weil divisors on non Noetherian schemes

Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...

**5**

votes

**1**answer

372 views

### Divisorial contraction: when is the image an algebraic space or a stack?

Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already ...

**5**

votes

**1**answer

541 views

### Stable base loci cannot contain isolated points

Let $X$ be a normal projective complex variety.
A theorem of Fujita-Zariski says that if $L$ is a Cartier divisor on $X$ such that
the base locus $Bs(|L|)$ is a finite set then $L$ is semiample.
It ...

**5**

votes

**1**answer

807 views

### Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...

**5**

votes

**1**answer

251 views

### Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...

**5**

votes

**1**answer

451 views

### Numerically equivalent effective divisors and semiampleness

Recall that a divisor $M$ on a variety $X$ is said to be semiample if $kM$ is base point free for a certain $k > 0$.
Being semiample is not a numerical property (take for example torsion and a ...

**5**

votes

**1**answer

287 views

### Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed.
Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...

**4**

votes

**7**answers

1k views

### Nef divisors with few global sections

Are there nef divisors D on a complex projective manifold X such that $h^0(X,D)$ is less than or equal to $\dim X$?
Edit: In fact I'm interested in nef line bundles D, not just divisors.

**4**

votes

**2**answers

713 views

### Special divisors on hyperelliptic curves

I was reading a proof that used the following result
Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree ...

**4**

votes

**2**answers

445 views

### Vague question on $Pic^0$

For a smooth variety $X$ when $Pic^0(X)$ is trivial, we get an isomorphism between $N^1(X)$ and Picard group and life become easier.
My question is whether in general there are theorems, criteria ... ...

**4**

votes

**2**answers

982 views

### Bertini's Theorem small print

Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ ...

**4**

votes

**2**answers

248 views

### A question on the effective cone

Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...

**4**

votes

**2**answers

392 views

### Varieties with big anti-canonical divisor

I recently heard about the following problem:
Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ?
Now, $-K_X$ big if and only if $-K_X ...

**4**

votes

**1**answer

204 views

### Extension of linear system

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on ...

**4**

votes

**1**answer

172 views

### Canonical bundle of moduli space of rational curves and automorphisms

Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves.
The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...

**4**

votes

**1**answer

592 views

### A working generalization of Weil divisors

Hartshorne defines Weil divisors under the hypotheses "Noetherian integral separated scheme regular in codimension 1", which, for example, ensures that the divisor of a rational function is a finite ...

**4**

votes

**1**answer

79 views

### Zeroes of global sections killed by differential operators

I asked this question some two weeks ago on StackExchange, but received no feedback of any sort ...
Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic ...

**3**

votes

**4**answers

810 views

### Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...

**3**

votes

**1**answer

425 views

### References about pseudoeffective cone

I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.

**3**

votes

**4**answers

461 views

### Ample divisors on $\mathbb{P}^3$ blow-up along single point

Let $\pi:X\to\mathbb{P}^3$ be the blowing up at single point with $E$ be the exceptional divisor. Let $H=\pi^\ast\mathcal{O}_{\mathbb{P}^3}(1)$.
In Ample divisors on the blow up of $\mathbb{P}^3$ at ...

**3**

votes

**3**answers

293 views

### Nefness of $h-e$ in the blowup of $\mathbb{P}^n$

Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor ...

**3**

votes

**2**answers

332 views

### Standard plane Cremona transformation

Let us consider nine general points $p_1,...,p_9\in\mathbb{P}^2$ and the line $L = \left\langle p_1,p_2\right\rangle$. Take the standard Cremona $f_1$ centred in $p_3,p_4,p_5$, then $C_1 = f_1(L)$ is ...