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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3
votes
1answer
98 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 $mult_xD < 1$ ...
0
votes
1answer
55 views

Global sections in negative degree line bundles over singular curves

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 ...
2
votes
1answer
119 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: ...
1
vote
2answers
174 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 ...
1
vote
1answer
94 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
181 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 ...
0
votes
1answer
111 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 ...
3
votes
0answers
106 views

Non-vanishing of $H^0$ on rational surface

Let $X$ be a smooth rational surface that admits a proper birational morphism to $\mathbb{P}^2$ and $D$ a simple normal crossing divisor on $X$ such that $K_X+D$ is big, is it true that ...
0
votes
1answer
115 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 $H$ ...
1
vote
1answer
133 views

Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points

Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let $$D = aH-b_1E_1-...-b_kE_k$$ be a divisor on $X$. Are there conditions on $a,b_1,...,b_k$ ensuring that ...
3
votes
1answer
162 views

A question about kawamata's proof of vanishing for big and nef $\mathbb{Q}$ divisors

Theorem 2 [1, p.46] Let $X$ be a non-singular projective algebraic variety of dimension $n$, and $D$ a numerically effective $\mathbb{Q}$-divisor such that $(D^n)>0$. We assume that the support of ...
0
votes
1answer
170 views

Simple normal crossing divisors

I found the following definition. A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple normal crossing if for every point $p \in X$ a local equation of $D$ is ...
-2
votes
1answer
124 views

Singular irreducible quadrics

Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by $$x_0^2+x_1^2+...+x_k^2 =0.$$ If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$. If ...
0
votes
3answers
266 views

Higher cohomology of sheaves on a projective space

Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf ...
4
votes
2answers
225 views

Varieties with big anti-canonical divisor

I recently heard about the following problem: Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ? Now, $-K_X$ big if and only if $-K_X ...
1
vote
2answers
246 views

Big and Nef divisors

In Example 2.2.19 of Lazarsfeld, Positivity in Algebraic Geometry I, I found the following statement: Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and ...
5
votes
1answer
400 views

Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
1
vote
2answers
99 views

Divisors with positive Iitaka dimension

Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$. Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ ...
5
votes
1answer
384 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 $p_1,...,p_n$. ...
5
votes
1answer
204 views

Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$. We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...
2
votes
4answers
254 views

Ample divisors on $\mathbb{P}^3$ blow-up along single point

Let $\pi:X\to\mathbb{P}^3$ be the blowing up at single point with $E$ be the exceptional divisor. Let $H=\pi^\ast\mathcal{O}_{\mathbb{P}^3}(1)$. In Ample divisors on the blow up of $\mathbb{P}^3$ at ...
1
vote
0answers
151 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
1answer
114 views

divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves

Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...
0
votes
0answers
139 views

Isomorphic Algebraic Geometric codes

Let $\chi/\Bbb F_q$ be an algebraic curve over a finite field $\Bbb F_q$ with $q^s$ rational points for some $s\in(0,1)$. Let $L(D)$ be the riemann roch space with prescribed zeros and poles. Let ...
4
votes
1answer
194 views

Extension of linear system

Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on ...
1
vote
1answer
263 views

Does every ample divisor “span” a hyperplane?

Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...
3
votes
0answers
124 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 ...
0
votes
0answers
47 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 ...
1
vote
0answers
53 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 ...
9
votes
2answers
198 views

Effectiveness of the distinguished theta characteristic in characteristic 2

Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic ...
-1
votes
1answer
126 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
218 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
243 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
214 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
235 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
169 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
67 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
146 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
78 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
118 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
328 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
415 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
425 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
179 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
141 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
299 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
720 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
274 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
177 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
273 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 ...