Questions tagged [divisors-algebraic-geometry]

For questions involving divisors, invertible sheaves and/or line bundles on varieties and schemes.

Filter by
Sorted by
Tagged with
5 votes
0 answers
70 views

Fibers of the Abel Jacobi map over curves

I am studying the Abel Jacobi map $$Div_{X/k} \to Pic_{X/k}$$ for projective, smooth, irreducible curve $X/k$ where $k$ is algebraically closed. Let $S=Spec(k)$, $T$ a curve over $k$ and let $\mathcal{...
user avatar
  • 51
1 vote
0 answers
20 views

Decomposing a big and nef divisor into ample + effective

Let $\pi: X \to \mathbf{P}^2$ be the blow-up of the projective plane at one point. Write $H$ for a hyperplane divisor of $\mathbf{P}^2$. The pullback $\pi^* H$ is big and nef, so it can be written in ...
user avatar
  • 93
0 votes
1 answer
57 views

Is a base point free nonspecial invertible sheaf generated by two global sections?

Let $X$ be a projective nonsingular integral curve of genus $g$ over an aglebraically closed field. In Hartshorne Chapter IV exercise 6.8, we know that: If $d\ge g+1$ then there is an effective ...
user avatar
  • 922
1 vote
1 answer
53 views

Calculating divisor of function on elliptic curve

I read Pairings for Beginners by Craig Costello. In the example 3.1.1 at 37-th page we consider $ E/F_{103} : y^2 = x^3 + 20x + 20$, with points $ P = (26, 20), Q = (63, 78), R = (59, 95), T = (77, 84)...
user avatar
  • 23
0 votes
0 answers
21 views

Can $\nu_Y (f) = 0$ for every prime divisor containing $Z = \overline{\{z\}}$, where $f \notin \mathcal{O}_z$?

This is about the argument in Hartshorne exercise III.6.8(a), which is supposed to show $X_s$ form a base for the topology of a noetherian, integral, separated, locally factorial scheme $X$, where $s$ ...
user avatar
  • 1,769
0 votes
0 answers
43 views

How is it that the quotitient of two rationals is a common divisor for both?

When $2$ number's ratio can not be expressed as a rational number then we call these number incommensurable. But I also read that incommensurable also has the meaning of being able to measure two ...
user avatar
  • 1,517
0 votes
0 answers
83 views

Why do we have $\mu_* \mathcal{O}_Y(K_{Y/X}) = \mathcal{O}_X$ for a resolution?

Let $X$ be a normal complex variety and $\mu : Y \to X$ a resolution of singularities. We define the relative canonical divisor to be $K_{Y/X} := K_Y - \mu^* K_X$. In his book Positivity in Algebraic ...
user avatar
  • 93
0 votes
0 answers
45 views

Why consider $dx/x$ on a complex curve?

In a paper I'm reading, the author considers a compact Riemann surface -- or smooth algebraic curve, you pick -- $X$ given by the equation $y^d=x^n-1$ for some natural numbers $n,d$. My understanding ...
user avatar
  • 1,719
2 votes
1 answer
68 views

What is a Cartier divisor on an affine scheme ring-theoretically?

A Cartier divisor is usually defined to be a section of the sheaf $\mathscr{K}^\times/\mathscr{O}^\times$. For an affine scheme, does a Cartier divisor on $\mathrm{Spec}(A)$ have a simple description ...
user avatar
  • 93
0 votes
1 answer
38 views

Definition of $\nu_C(f)$ where $C$ is a subvariety of codimension 1 and $f$ is rational.

I am studying divisors from Shafarevich. The book is available as a free download at http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf. My issue comes at page 164 of the pdf. Assume we ...
user avatar
0 votes
1 answer
50 views

Is a degree zero divisor on a curve always basepoint-free?

Let $X$ be a smooth projective curve and $D$ a divisor on $X$ of degree zero. Is it always the case that $D$ is basepoint-free? If not, then is there always some (positive) power $mD$ which is?
user avatar
  • 93
3 votes
1 answer
104 views

Grothendieck Group of a Nonsingular Curve (Hartshorne Exercise II.6.11).

I have copied the exercise below for reference. I was able to figure out how to do (a) and (d), so let me focus on (b) and (c). Please do not provide me with full solutions, but hints (and if ...
user avatar
0 votes
0 answers
41 views

Nef divisor on a surface and intersection number

Let $X$ be a complex surface and $D$ a nef divisor that is not numerically trivial. Then for any $n\in \Bbb N$, can we choose a smooth curve $C$ such that $D\cdot C\geq n$? Certainly there is a ...
user avatar
  • 2,034
0 votes
0 answers
48 views

On a particular exact sequence in cohomology

The setup for my question is as follows (from page 9 of Deschamps' expository notes on the Artin-Winters proof of semi-stable reduction here). We want to prove that for $X$ the special fiber of a ...
user avatar
1 vote
0 answers
23 views

$h^0(S^nF(nD_1+D_2))=O(n^2)$ for a rank 2 vector bundle $F$ on a smooth curve

The following proposition and proof are given in Lemma 2.5 of https://mathscinet.ams.org/mathscinet-getitem?mr=1272710, and I have some questions about it. Proposition. Let $F$ be a rank 2 vector ...
user avatar
  • 2,034
0 votes
0 answers
59 views

Applying Riemann-Roch to a rank 2 vector bundle on a complex surface

Let $X$ be a complex surface, $F$ a rank 2 vector bundle on $X$, $n$ a positive integer, $S^nF$ the $n$-th symmetric product of $F$, $P$ a rational divisor, and $a$ a rational number such that $naP$ ...
user avatar
  • 2,034
0 votes
1 answer
38 views

Projective surface has an ample divisor

Let $S$ be a projective surface. In https://mathoverflow.net/questions/63999/nef-divisor-on-surface, there is an argument proving that any nef divisor $D$ on $S$ has $D^2\geq 0$. But the argument ...
user avatar
  • 2,034
2 votes
0 answers
74 views

About quotient singularities on a complex projective surface

Let $S$ be a normal (complex) projective surface with quotient singularities (locally analytically isomorphic to $\Bbb C^2/G$ where $G\subset \text{GL}(2,\Bbb C)$ is a finite group whose action is ...
user avatar
  • 2,034
1 vote
1 answer
27 views

Either $\pm c_1(S)$ is ample or $c_1(S)=0$ for a normal projective surface with quotient singularities with $b_2(S)=1$

According to this paper: https://arxiv.org/pdf/math/0602562.pdf, in p.2 (below Theorem 1), it is written that if $S$ is a normal projective surface (so there are only finitely many isolated ...
user avatar
  • 2,034
1 vote
1 answer
61 views

Canonical effective divisors on a degree 4 curve

I am trying the following problem from Hartshorne: Original Problem (IV.3.2.i): Let $X$ be a plane curve of degree 4. Show that the effective canonical divisors on $X$ are exactly the divisors $X \...
user avatar
0 votes
1 answer
69 views

Sections of pullback bundle

Let $X$ be a genus 3 curve canonically embedded in $\mathbb{CP}^2$. Why is it that the line bundle $L$ obtained by pulling back the hyperplane bundle $\mathcal{O}_{\mathbb{P}^2}(1)$ has 3 independent ...
user avatar
  • 33
0 votes
0 answers
35 views

Embedding of complex torus into $\mathbb{P}^3$

I'm dealing with Riemann Surfaces and I saw how a basis for the space $L(D)$ (where $D\in Div(X)$ is a divisor for the complex torus $T=\mathbb{C}/\Lambda$) of D-bounded meromorphic functions give ...
user avatar
  • 87
3 votes
1 answer
70 views

What is a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective?

I am trying to solve an exercise that asks for a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective. I think I have mostly solved it but since I am a bit shaky ...
user avatar
  • 33
0 votes
1 answer
81 views

How to define the intersection product of a curve with itself: $(C\cdot C)$?

I'm reading through Positivity in Algebraic Geometry by Lazarsfeld, and it defines the intersection product for bundles $L$ and (Cartier) divisors $D$ on a complete, irreducible complex variety $X$, ...
user avatar
  • 1,482
1 vote
1 answer
40 views

Rank of normal sheaf in singular varieties

Let $X$ be a $n$-dimensional normal, singular projective variety (over the field of complex numbers), and let $Y$ be a subvariety of $X$ of dimension $k$. I know that, if $X$ was non-singular, than ...
user avatar
  • 23
2 votes
1 answer
66 views

Possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$

I am looking for the possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$. By combining the Brill-Noether theorem with the equality $g={d-1\choose 2}$ for a plane ...
user avatar
  • 1,482
2 votes
0 answers
48 views

$D-P-Q$ is nonspecial for all $P, Q \in X$ implies $D$ is nonspecial

(Sorry for my bad English) For the proof of Halphen's theorem, in Hartshorne p.349 I need this: A divisor $D$ on a curve $X$ is nonspecial and very ample if and only if $D-P-Q$ is nonspecial for all $...
user avatar
  • 83
2 votes
1 answer
39 views

Hom set between curves on a surface

Let $S$ be a smooth projective surface and $i:C\subset S,j:D\subset S$ be two smooth curves on $S$ intersecting each other transversely. How can we say about the set $\text{Hom}_S(i_*\mathcal{O}_C,j_*\...
user avatar
1 vote
1 answer
94 views

Exact sequence in Hartshorne's proof of Clifford's theorem (Theorem IV.5.4)

In Theorem 5.4 (Clifford) of chapter IV of Hartshorne we have an exact sequence: $$0\rightarrow \mathscr{L}(D')\rightarrow \mathscr{L}(D)\oplus \mathscr{L}(E)\rightarrow \mathscr{L}(D+E-D')\rightarrow ...
user avatar
  • 83
3 votes
0 answers
76 views

What does linear equivalence geometrically mean for varieties?

Suppose we have a sufficiently nice scheme or say we are working with an abstract nonsingular variety $X$ over an algebraically close field. In this setting, one can study (Weil) divisors. It is then ...
user avatar
4 votes
0 answers
59 views

First Chern class coincides with degree of divisor without poincare duality or de rham cohomology

I know there are a lot of references (e.g. Griffiths-Harris page 141), but the issue is that these references always prove the proposition in arbitrary dimensions, using a somewhat contrived ...
user avatar
  • 351
1 vote
1 answer
78 views

Can effective divisors generate all Cartier divisors?

Let $X$ be a Noetherian scheme. My question is that: for any Cartier divisor $D$, can we write it as $D_1-D_2$ where $D_1$,$D_2$ are effective? What about further assume $X$ is integral? I can see $D$ ...
user avatar
  • 922
0 votes
0 answers
43 views

General way of determine where a rational function $\phi\in k(X)$ is not regular

So, if you have an irreducible variety $X$ over $k=\bar{k}$ and you consider a rational function $\phi=f/g\in k(X)$, is there a general way to determine where $\phi$ is not regular? At this point I ...
user avatar
  • 148
0 votes
1 answer
53 views

Are all Weil divisors principal?

I've been studying Weil divisors from Vakil's FOAG Chapter 14.2. I'm somewhat confused, as to why every Weil divisor on a Noetherian normal irreducible scheme $X$ needn't be principal. My reasoning is ...
user avatar
0 votes
1 answer
43 views

Confusion about sum of ramification indices

In algebraic number theory, I'm well aware of the following formula: Given the "$AKLB$ setup" and a prime $\mathfrak{p}$ of $A$, then \begin{equation}\sum_{\mathfrak{P}|\mathfrak{p}} e_{\...
user avatar
  • 1,208
1 vote
1 answer
78 views

Effective divisor on smooth projective variety

Let $X$ be a smooth projective variety and $D_0$ be a divisor on $X$ (over the algebraically closed field $K$). Then according to Harthshorne Proposition $7.7(b)$(Chapter $2$) every effective divisor ...
user avatar
  • 149
2 votes
3 answers
119 views

Proof that for an effective divisor $l(D)\leq \deg D +1$

In one of my recent questions I got this answer: https://math.stackexchange.com/a/4348876/645867. In particular, it is said that "for an effective divisor $l(D)\leq \deg D +1$ and equality holds ...
user avatar
  • 177
2 votes
1 answer
86 views

Possible values of $l(np)$ for $p\in X$, where $X$ is a compact Riemann surface of genus 2

Here $l(D)$ denotes the dimension of the complex vector space $$\mathcal{L}(D)=\{f\colon X \to \mathbb{P}^1\mathbb{C} \hspace{5pt} \mathrm{meromorphic} \hspace{5pt} \mathrm{s.t.} \hspace{5pt} (f)+D \...
user avatar
  • 177
3 votes
0 answers
62 views

Relative Cartier divisor scheme

This is question 1.14.1.2 in Kollar's book Rational Curves on Algebraic Varieties. Let $f:X\rightarrow S$ be a flat and projective morphism with integral fibres such that $H^1(X_s,O_{X_S})=0$ for ...
user avatar
  • 681
1 vote
0 answers
49 views

Cusp in the cuspidal cubic as a non effective Cartier Divisor

I have a basic question about the cusp in the cuspidal cubic. Take $X$ to be $y^2=x^3$ in $\mathbb{A}^2$ (over an algebraically closed field for instance), it is well known that the point $(0,0)$ ...
user avatar
  • 121
0 votes
1 answer
79 views

Doubts about proof of Castelnuovo theorem IV, 6.4 of Hartshorne

In the proof of Theorem 6.4 (Castelnuovo) of Hartshorne, chapter IV, it reads: "to show that $P_i$ is not a base point of the linear system $|nD-P_1-\dots-P_{i-1}|$ it is sufficient to find a ...
user avatar
  • 109
2 votes
0 answers
106 views

A Hyperelliptic Compact Riemann Surface $X$ is a 2-sheeted cover of $\mathbb{P}^1$

It is stated in Geometry of Algebraic Curves by Harris that there are two equivalent definitions of a hyperelliptic curve $X$: (i) There exists a degree $2$ meromorphic global function of $X$. (ii) $X$...
user avatar
  • 1,482
0 votes
1 answer
53 views

Line bundle of point is twisting sheaf of Serre on projective line

I think I misunderstood something fundamental, but I cannot figure out where. Let $X = \mathbb{P}^1$ and let $D$ be the principal divisor $[0:1]$. I believe that $\mathcal{L}(D) = \mathcal{O}(1)$, ...
user avatar
  • 351
4 votes
1 answer
62 views

A finite correspondence in $Cor_k(X,\mathbb A^1)$ is a principal Weil divisor

I am trying to understand Lemma 4.4 of "Lectures on Motivic Cohomology" by Mazza-Weibel-Voevodsky. In the following, $X$ is a smooth scheme over a field $k$. In the proof of the lemma, the ...
user avatar
  • 3,902
0 votes
0 answers
39 views

Proving that a rational function is a section of an invertible sheaf

Let $S$ be a surface, $E$ be a curve on $S$, and $H$ be a hyperplane section of $S$. Let $a\in H^0(S,\mathcal O_S(H+(k-1)E))$, and $b\in H^0(S,\mathcal O_S(H+kE))$. Let $U$ be an open subset of $S$ on ...
user avatar
  • 87
3 votes
0 answers
57 views

A combinatorical approach to classical Riemann-Roch

I am reading "Riemann-Roch and Abel-Jacobi Theory on a Finite Graph" by Baker and Norine (2007, arXiv 0608360). In this paper, the authors formulate abstract criterions for a set X, its set ...
user avatar
  • 31
1 vote
1 answer
35 views

Embeddings of a surface that are given by linear systems of divisors

I am reading the proof of Castelnuovo's contractibility criterion in Beauville's Complex Algebraic Surfaces. I would like to clarify a paragraph. We have a hypeplane section $H$ of a surface $S$, a ...
user avatar
  • 87
0 votes
1 answer
36 views

Embedding of a variety

Let $D$ be a divisor on a variety $X$, and assume that $h^0(X,\mathcal O_X(D))=n+1$. So let $s_0,\dots,s_n$ be a basis of $H^0(X,\mathcal O_X(D))$. I am trying to understand the following statement : &...
user avatar
  • 87
1 vote
0 answers
71 views

Hartshorne Exercise II 6.11 (c): Grothendieck group of a nonsingular curve

Exercise: Let $X$ be a nonsingular curve over an algebraically closed field $k$. (c) If ${\mathscr{F}}$ is any coherent sheaf of rank $r$(means that its stalk at the generic point has dimension $r$ ...
user avatar
  • 922
0 votes
0 answers
83 views

Existence of a certain polynomial on $\mathbb P_k^n$

Let $k$ be an infinite field. Let $X\subseteq\mathbb P_k^n$ be a projective subvariety. Let $D$ be a Cartier divisor on $X$. Mumford and Oda’s Algebraic Geometry II says (in the paragraph following ...
user avatar
  • 482

1
2 3 4 5
9