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
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1answer
105 views

Divisor on variety determined by its restriction to curves

Is a (Cartier) divisor on a variety uniquely determined by its restriction to curves inside the variety? If so, how do we see this?
2
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1answer
162 views

A necessary condition for existence of Ricci flat metric on pair (X,D)

Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K_X +D = 0$ necessary for the existence of Ricci-flat metric?
10
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1answer
257 views

Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?

Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map $$ \phi: X(\mathbb C) \to \mathbb P^...
3
votes
0answers
59 views

Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...
2
votes
1answer
82 views

Linear systems and 2-torsion shifts on hyperelliptic curves

Let $C$ be a hyperelliptic curve of genus $g$ and let $D$ be a divisor on $C$ of degree $g+1$. Assume that the linear system $|D|$ is base-point-free. Now add a $2$-torsion point $[E]$ to $D$. I would ...
3
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1answer
91 views

Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points. Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...
4
votes
1answer
80 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 ...
14
votes
1answer
330 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 ...
6
votes
1answer
190 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 $D^{2}=0$?
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0answers
56 views

Polarization of the Prym variety

Let $X\rightarrow Y$ a ramifield double cover of curves, $J_X, J_Y$ their jacobians, $P\subset J_X$ the Prym variety, for any line bundle $L$ on $X$ of degree $g_X-1$, denote by $\Theta_L$ the ...
2
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0answers
134 views

Ricci flat metric on pair (X,D)

Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...
2
votes
1answer
158 views

Inverse image of a divisor

Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor. Let us look at the cycle $f^{-1}(D)\subset X$...
1
vote
1answer
140 views

Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$. Now, let $f:X\...
0
votes
2answers
172 views

Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim? [closed]

Edit 01:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, ...
7
votes
2answers
505 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 ...
6
votes
1answer
514 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 $...
1
vote
1answer
299 views

Negative degree line bundles over a singular projective curve have no sections?

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
3
votes
1answer
166 views

Question about canonical DM stacks

Let $\mathcal{X}$ be a canonical stack (edit: I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. ...
1
vote
1answer
244 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 $...
1
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0answers
181 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. ...
2
votes
0answers
108 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 trivial?...
0
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1answer
83 views

Rational functions on hyperelliptic Riemann surface

Let $\mathcal R$ be an hyperelliptic Riemann surface of genus $g\geq 1$. Is it true that the only possible rational functions on $\mathcal R$ with $\leq g$ poles are the liftings of rational functions ...
1
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1answer
161 views

Tor-amplitude [0, 1] in the setting of intersection theory on a regular surface?

The question is coming from Definition 1.5 in Deligne's Expose X in SGA 7 on intersection theory. Let $X$ be a connected regular scheme of dimension $2$ and $Y \subset X$ a reduced divisor that ...
0
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0answers
97 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 ...
1
vote
1answer
264 views

symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?
9
votes
2answers
323 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 ...
1
vote
1answer
312 views

On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor. People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of $...
9
votes
1answer
466 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 \...
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0answers
72 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 ...
4
votes
2answers
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 particular,...
2
votes
1answer
335 views

A question about an intersection number

Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
2
votes
1answer
193 views

Reference request: log Fano varieties

I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano. Thanks a lot.
1
vote
1answer
150 views

Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor?

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...
0
votes
1answer
134 views

Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
0
votes
1answer
143 views

Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...
1
vote
1answer
267 views

Moving a divisor on a (reducible, non-reduced) curve

I am trying to understand the first sentence of the proof of 9.1/5 in "Neron models." There we have a proper curve $X$ over a field $K$ and a line bundle $\mathscr{L}$ on $X$. Our ultimate goal is to ...
6
votes
4answers
911 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 = \mathcal{O}_X(...
2
votes
3answers
661 views

Movable Divisors

Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?
11
votes
2answers
398 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 $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$. ...
7
votes
1answer
559 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 $p_1,...,p_n$. ...
3
votes
2answers
335 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 ...
1
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0answers
172 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 ...
4
votes
1answer
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 ...
3
votes
1answer
164 views

A question on klt pairs

Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
2
votes
1answer
283 views

Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper: http://ac.els-cdn.com/...
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2answers
239 views

Small birational maps and singularities of the pair

Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\...
1
vote
1answer
130 views

Boundedness of the number of curves negative on a varying big divisor

For a divisor $D$ on a smooth complex projective surface $X$, the stable fixed part is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the ...
1
vote
1answer
225 views

Cremona transformations

Let $f:\mathbb{P}^n_1\dashrightarrow\mathbb{P}^n_2$ be the standard Cremona transformation based on $p_1,...,p_{n+1}\in\mathbb{P}^n_1$ and $q_1,...,q_{n+1}\in\mathbb{P}^n_2$. That is, $f$ is the ...
1
vote
1answer
141 views

Big divisors and small transformations

Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that: $-K_Y$ is ...
0
votes
1answer
134 views

Dimension of image of a hyperplane section

If we have a surjective morphism $f:X\to Y$, where $X$ is $n$ dimensional projective variety and $Y$ is $m$ dimensional projective variety. If $m<n$, Can we choose a general hyperplane section $H$ ...