**1**

vote

**1**answer

221 views

### Degree of the negative part of a divisor

Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for a more precise condition). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree ...

**0**

votes

**1**answer

160 views

### Volume form on pair (X,D)

Let $X$ be a singular Kahler variety with Kahler current $\omega $ then the volume form is $\omega^n$. Now let $D$ be a divisor then how can we define volume form on pair $(X,D) $?

**12**

votes

**0**answers

716 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 ...

**8**

votes

**0**answers

941 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 ...

**3**

votes

**0**answers

169 views

### Hypersurfaces with Gorenstein singular loci

Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...

**3**

votes

**0**answers

286 views

### On generators of the Picard group of a projective smooth surface over a finite field

Let $X$ be a smooth projective surface over a finite field $k=\mathbb{F}_q$. Let us first review the proof of the finite generation of $Pic(X)$ (notice that the proof is valid for any smooth ...

**2**

votes

**0**answers

93 views

### On triviality and numerical triviality of (classes of) divisors

Let $X$ be a smooth irreducible threefold, and let $H$ be an ample divisor on $X$.
Assume that $D$ is a divisor on $X$ such that $D\cdot H^2=D^2\cdot H=D^3=0$.
Question 1: Is $D$ numerically ...

**2**

votes

**0**answers

145 views

### A nice way to verify whether the Neron-Severi group of a smooth affine variety is zero

Let $S$ be a smooth affine variety over an algebraically closed field (this could be the field of complex numbers). Is there an 'easy' way to verify whether $NS(S)=0$? Unfortunately, I don't know how ...

**2**

votes

**0**answers

512 views

### Riemann-Roch for ARBITRARY Function Fields

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. ...

**2**

votes

**0**answers

333 views

### A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...

**2**

votes

**0**answers

244 views

### Proof of Saito criterion

Does it exist another proof of saito's criterion for free divisors, other than the one in "K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo ...

**1**

vote

**0**answers

101 views

### Family $(X_y,D_y)$ with trivial canonical bundles

Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...

**1**

vote

**0**answers

133 views

### A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...

**1**

vote

**0**answers

190 views

### Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...

**1**

vote

**0**answers

62 views

### lift sections on a thickened curve

Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X.
Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...

**1**

vote

**0**answers

145 views

### cohomology of a normal crossing divisor

Let $D$ be a simple normal crossing divisor on a smooth projective variety over a field $k \subset \mathcal{C}.$ Write $D_i$ with $i \in I$ for its irreducible components. Denote, as usual,
...

**1**

vote

**0**answers

293 views

### Canonical divisor of a curve base point free (if g>0)

Is there a way to prove that the canonical divisor $W$ of an algebraic function field in one variable $F$ over a field $K$ (that is the function field of an algebraic curve) of genus $g>0$ is base ...

**0**

votes

**0**answers

162 views

### Bigness of a symplectic form on pair $(X,D)$

Let $(M,\omega_M)$ be a compact Kähler manifold. We say that a semi-positive $(1,1)$ form $\omega$ is big iff $$\int_M\omega^n>0$$.
Now let we have the pair $(X,D)$ where $D$ is a divisor on ...

**0**

votes

**0**answers

89 views

### Universal property of complete linear systems

Let $X$ be a projective scheme over a field $k$ and $S$ a $k$-scheme. Fix a closed immersion $i:X \to \mathbb{P}^n$ for some $n$ and denote by $\mathcal{O}_X(1):=i^*\mathcal{O}_{\mathbb{P}^n}(1)$. Let ...

**0**

votes

**0**answers

69 views

### Terminology regarding divisor on a curve

Suppose that $D = \sum n_i P_i$ is a divisor on a curve $C$, say, over a field. Is there a standard algebraic geometry terminology referring to the set $\{ P_i : n_i \neq 0 \} \subset |C|$? Support of ...

**0**

votes

**0**answers

52 views

### open subset in constructible set of divisors

Let a smooth projective curve $X$ over $\mathbb{C}$.
Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$.
Let $N=\deg (D)$ and ...

**0**

votes

**0**answers

75 views

### sections of vector bundles transversal to a divisor

Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$.
$E$ a vector bundle over $X$ with a divisor $D$.
We look at the sections $A:=H^{0}(X,E)$ with $\deg E$ big enough.
...

**0**

votes

**0**answers

80 views

### invertible sheaf of a hypersurface

Let $X \hookrightarrow \mathbb{P}^n$ be a hypersurface of degree $d$. I am trying to prove that $\mathcal{O}_{\mathbb{P}^n}(X)=\mathcal{O}(d)$. My idea is the following: if one considers the $d$-uple ...

**0**

votes

**0**answers

103 views

### How to compute the Betti numbers of S-D for a surface S and a divisor D?

Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper ...

**0**

votes

**0**answers

221 views

### Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...