Questions tagged [divisors-algebraic-geometry]
For questions involving divisors, invertible sheaves and/or line bundles on varieties and schemes.
445
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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{...
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0
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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 ...
- 93
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1
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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 ...
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1
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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)...
- 23
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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 $s$ ...
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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 ...
- 1,517
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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 ...
- 93
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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 ...
- 1,719
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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 ...
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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 ...
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1
answer
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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?
- 93
3
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1
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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 ...
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41
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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 ...
- 2,034
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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 ...
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$h^0(S^nF(nD_1+D_2))=O(n^2)$ 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 ...
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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 $naP$ ...
- 2,034
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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 ...
- 2,034
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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
...
- 2,034
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1
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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 ...
- 2,034
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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 \...
- 541
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1
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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 ...
- 33
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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 ...
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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 ...
- 33
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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 $X$, ...
- 1,482
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1
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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 ...
- 23
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1
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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 ...
- 1,482
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$D-P-Q$ 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 $...
- 83
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1
answer
39
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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_*\...
user999577
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1
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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 ...
- 83
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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 ...
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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 ...
- 351
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1
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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$ ...
- 922
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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 ...
- 148
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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 ...
- 465
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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_{\...
- 1,208
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78
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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 ...
- 149
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3
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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 ...
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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 \...
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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 ...
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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 $(0,0)$ ...
- 121
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1
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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 ...
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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) $X$...
- 1,482
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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 $\mathcal{L}(D) = \mathcal{O}(1)$, ...
- 351
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1
answer
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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 ...
- 3,902
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0
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39
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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 ...
- 87
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0
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57
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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 ...
- 31
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1
answer
35
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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 ...
- 87
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votes
1
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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 :
&...
- 87
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0
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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 $r$ ...
- 922
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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 ...
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