For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.

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-1
votes
1answer
139 views

effective divisors on a curve and upper semi-continuity

Let consider a smooth projective curve $X$ over $\mathbb{C}$. We consider the scheme that classifies effective divisors of degree $d$, which is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the ...
1
vote
1answer
228 views

a question on the space of divisors on a curve

Let $X$ a complex curve and $x\in X$ a point. We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group. ...
0
votes
2answers
261 views

“rationality” of divisors

Let $X$ be a smooth projective variety over some field $k$. Then each closed point $x$ has an associated residue field $k(x)$ which is a finite extension of $k$ and a point is rational when $k(x)=k$. ...
2
votes
1answer
314 views

what are the singularities of a normal crossings divisor?

This is probably a very stupid question. I'm sorry. Let $D$ be a simple normal crossings divisor on some smooth projective variety $D$. By this I mean that the irreducible components $D_i$ are smooth ...
1
vote
1answer
243 views

a question on Euler characteristic of normal crossing divisors

Let $X$ be a smooth, projective complex algebraic variety. Let $D$ be a simple normal crossings divisor on $X$, with irreducible components $D_i$, for $i \in I$. For each non-empty subset $J \subset ...
2
votes
1answer
183 views

Conical divisor over a $\mathbb Q$-Cartier divisor.

I would like to know if the following statement is correct. Statement. Let $X$ be a normal projective variety with $Pic(X)=\mathbb Z+torsion$. Let $L$ be an ample line bundle on $X$ and let $D$ be an ...
0
votes
0answers
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. ...
1
vote
1answer
190 views

On divisorial correspondences between curves

Assume we are given two smooth curves $C_1$ and $C_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms between their ...
0
votes
0answers
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 ...
1
vote
0answers
141 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, ...
2
votes
1answer
415 views

On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
1
vote
2answers
579 views

Embedded resolution of singularities

I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities". Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" ...
7
votes
2answers
468 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 ...
2
votes
1answer
190 views

divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...
2
votes
0answers
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 ...
3
votes
1answer
319 views

Roots of line bundles

Let $k$ be a fixed algebraically closed field and $X/k$ an irreducible scheme smooth and proper over $k$. Can there exist a line bundles $\mathcal{L}, \mathcal{M}$ and an integer $m > 0$ so that ...
3
votes
2answers
829 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
1answer
318 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 ...
1
vote
1answer
180 views

The pseudoeffective cone does not contain lines

It seems to be well-known that the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a normal variety $X$ does not contain lines through the origin. How can it be proved? Is there a reference?
2
votes
1answer
283 views

Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d? e.g. for $d=4$ the cohomology ...
2
votes
1answer
199 views

Constructing rational functions with ramification locus the divisor of some $n$-form

I'm still busy learning the theory of linear systems for compact Riemann surfaces. If the answer to the following question is negative, then there might not be any point in continuing. Let $X$ be a ...
0
votes
0answers
102 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 ...
5
votes
1answer
423 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 ...
2
votes
1answer
289 views

Do divisors of degree g with this property exist in general

I have the following question. It's a long shot, but worth the try. Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ ...
3
votes
2answers
288 views

When is the Wendt binomial circulant determinant divisible by 3?

The Wendt binomial circulant determinant $W_n$ can be defined quite simply as a resultant: $$ W_n = \operatorname{res}(x^n-1, (x+1)^n-1). $$ Truer to its name, one may also define it as the ...
2
votes
1answer
199 views

Divisor intersecting non-negatively the negative part of its Zariski decomposition

Hi all. I'm looking for an example of a smooth projective surface $X$ and a pseudo-effective divisor $D$ on $X$ such that when I consider the Zariski decomposition $D=P+N$ there is some component $E$ ...
1
vote
0answers
286 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 ...
3
votes
1answer
422 views

Is every Weil divisor on an arithmetic surface Q-Cartier

This question is about a technical issue I ran into. Let $S$ be a connected 1-dimensional Dedekind scheme, and let $X\to S$ be a flat projective integral normal 2-dimensional scheme. (For simplicity, ...
5
votes
3answers
786 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 ...
1
vote
2answers
328 views

Top self-intersection of the tautological line bundle

Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection ...
8
votes
0answers
916 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
1answer
382 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 $X$.
0
votes
0answers
219 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 ...
3
votes
2answers
774 views

Ample divisors on blown-up projective space

Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...
2
votes
1answer
559 views

Self intersection of blown up points and the lines which they lie on

I'm currently trying to understand the process of blowing-up, and a few things strike me as a little difficult to get an intuitive understanding of what's happening. The current problem is on self ...
3
votes
1answer
161 views

Boundedness of $C.K$ on a surface with $-K$ pseudoeffective

Let $S$ be a projective surface with pseudoeffective anticanonical divisor $-K_S$. Is it true that if $C$ is an integral curve with $C^2<0$ and $C \cdot K_S >0$, then $\max_C (C \cdot K_S)$ is ...
3
votes
2answers
974 views

(Anti)Canonical divisor of a blow up

This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky. Let X be a rational surface obtained by n ...
2
votes
2answers
1k views

Generalisations of Riemann-Roch for surfaces

Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have $$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$ This is the famous ...
3
votes
2answers
322 views

Moving a canonical divisor on a normal surface away from the singular locus

In a previous question Moving a Weil divisor on a normal surface away from a finite set of closed points I probably asked for too much. As J.C. Ottem pointed out, it is not always possible to move a ...
2
votes
3answers
405 views

Moving a Weil divisor on a normal surface away from a finite set of closed points

Let $Y$ be a normal surface and let $X$ be a closed subscheme of codimension 2, i.e., $X$ is a finite set of closed points. Let $D$ be a Weil divisor on $Y$. Question. Does there exist a Weil ...
2
votes
1answer
280 views

resolution of singularities and a projection formula

Let $Y$ be a normal surface and let $p:X\longrightarrow Y$ be a resolution of singularities. Let $f$ be a rational function on $Y$. Do we have that $p_\ast$div $(d(f\circ p)) = $ div $df$ as ...
2
votes
2answers
312 views

Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...
29
votes
10answers
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 $2$. ...
2
votes
1answer
1k views

Cone of effective divisors!

Let $X$ be a smooth simply connected projective variety of dimension $n$ (over complex numbers of course). For such $X$ we have two famous cones which are cone of effective curves and ample cone and ...
15
votes
2answers
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 ...
3
votes
0answers
284 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
2answers
690 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 ...
13
votes
2answers
2k 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 ...
4
votes
2answers
432 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 ... ...
8
votes
1answer
426 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 ...