All Questions
Tagged with divisors algebraic-surfaces
26 questions
27
votes
5
answers
7k
views
blowing up, -1 curves, effective and ample divisors
Lets say we're on a smooth surface, and we blow up at a point.
Is there a simple explicit computation that shows to me the fact that the exceptional divisor E has self intersection -1 ? I don't ...
- 271
14
votes
0
answers
3k
views
Ample divisors on projective surfaces
Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$?
Background: I was reading ...
- 5,359
7
votes
1
answer
543
views
A constructive proof of the theorem of the cube
Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
- 1,833
6
votes
3
answers
1k
views
Are there (-2)-curves on an Enriques surface?
Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it ...
- 427
6
votes
2
answers
2k
views
Generalisations of Riemann-Roch for surfaces
Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...
- 2,215
6
votes
2
answers
422
views
Nef divisors on surfaces
Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties:
$...
- 8,998
5
votes
1
answer
870
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 interesting)....
- 7,722
5
votes
1
answer
535
views
Volume of a divisor on a smooth projective surface
Let $X$ be a smooth projective surface (over complex numbers). Let $D$ be a divisor on $X$. Then we know that its volume is defined as $$\text{vol}_X(D):= \lim \sup_{m \rightarrow \infty} \frac{h^0(X,...
- 267
5
votes
1
answer
304
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 ...
- 761
4
votes
2
answers
2k
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 $ ...
- 2,215
4
votes
2
answers
1k
views
Isolated conics on a del Pezzo surface
Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated ...
- 1,488
4
votes
1
answer
169
views
Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve
Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see
Lange and Ruppert - Complete ...
- 1,833
4
votes
0
answers
172
views
How close is $h^0(mD)$ to be a polynomial?
Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies.
At ...
- 625
3
votes
2
answers
396
views
Abelian varieties corresponding to Hodge substructures
In an exercise of Voisin book, says:
Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set
$H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$.
...
- 519
3
votes
1
answer
228
views
Boundedness of $C.K$ on a surface with $-K$ pseudoeffective
Let $S$ be a projective surface with pseudoeffective anticanonical divisor $-K_S$. Is it true that if $C$ is an integral curve with $C^2<0$ and $C \cdot K_S >0$, then $\max_C (C \cdot K_S)$ is ...
- 427
2
votes
1
answer
411
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 ...
- 2,232
2
votes
2
answers
1k
views
Global sections of a linear system
Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...
- 2,215
2
votes
1
answer
464
views
Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?
I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely:
let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
- 115
2
votes
0
answers
149
views
Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces
We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
- 41
2
votes
0
answers
154
views
Structure of the big cone and Seshadri constant on Fano manifolds
I would like to know something about the following two questions.
Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define
\begin{gather}
\sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\,...
- 71
1
vote
1
answer
183
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 ...
- 13.8k
1
vote
0
answers
47
views
Positivity of self-intersection of dicisor associated to meromorphic function
In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic ...
- 111
1
vote
0
answers
168
views
Rational classes of $(-2)$-curves in a minimal surface of general type
Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
- 41
1
vote
0
answers
91
views
Picard numbers of isogenous K3 surfaces over a non-closed field
Let $S_1, S_2$ be K3 surfaces defined over a field $k$ and $\phi\!: S_1 \dashrightarrow S_2$ a dominant rational $k$-map (so-called isogeny). It is known that $\rho(S_1) = \rho(S_2)$ for the complex ...
- 1,833
0
votes
1
answer
271
views
Pseudoeffective divisors on surfaces
Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
- 8,998
-1
votes
1
answer
895
views
Restriction of a Cartier divisor
Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a closed ...
- 5,986