0
votes
0answers
138 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q [migrated]

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
4
votes
1answer
111 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
104 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
67 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
126 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
180 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
98 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
184 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
114 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
116 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
137 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
166 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
204 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
128 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
275 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
232 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
277 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
429 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
105 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
385 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
207 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
269 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
155 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
141 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
195 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
266 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
130 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
55 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
201 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
219 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
246 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
237 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
236 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
149 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
122 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
333 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
433 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
427 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
300 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
733 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
278 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 ...