Tagged Questions

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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Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$. Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
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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 ...
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Bertini theorems for base-point-free linear systems in positive characteristics

Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
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Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all. My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
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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 ...
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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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Cone of movable curves

Let $X$ be a smooth complex projective variety of dimension $n$. Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure ...
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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 $\mathbb{P}^n$...
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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 ...
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A working generalization of Weil divisors

Hartshorne defines Weil divisors under the hypotheses "Noetherian integral separated scheme regular in codimension 1", which, for example, ensures that the divisor of a rational function is a finite ...
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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 ...
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Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
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I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.
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 ...
Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...