Choosing a general set of $n$ points on a curve of genus at least $n$, you can assume that the divisor they define has a unique global section. Let $P$ denote the poset generated by all the possible ...

Let $G$ denote the group of rotations of the plane fixing the set of rational points. This group has multiple "names": it is $SO(2,\mathbb{Q})$, as Donu already commented; it is the group $\mathbb{...

Just to complement quim's answer, note that if $S$ is defined over the algebraic closure of a finite field, then the anticanonical divisor of the blow up of 9 general points in $\mathbb{P}^2$ is ...

Let $-K=N+E$ be the Zariski decomposition of $-K$, so that $N,E$ are $\mathbb{Q}$-divisors with $N$ nef, $E$ effective, such that $N \cdot E = 0$ and the restriction of the intersection pairing to the ...

Assume that $p$ is non-zero. If the form $dp/p$ were exact, then locally a primitive would be $log(p)+const$; this is easily seen not to work as soon as you can "loop around" $S$ (e.g. restrict ...

Asking individual vertices, you figure out the valence of each vertex with $n$ questions. Asking for pairs of vertices, you can then decide if each pair has an edge between them or not: they have an ...

Per Martin's request, here is a more detailed version of my comment. Given a graph $G$, the independence game is the game in which two players take turn in removing a vertex and all of its neighbors. ...

I do not know a reference, but the following short argument seems to work. Assume that the dimension of $X$ is at least 1! Argue by induction on the dimension of $X$. Reduce to the case in which ...

I am not too familiar with algebraic spaces, but maybe this is an example of an algebraic space that cannot be a quotient of a scheme by a finite group. The property of such quotients that I use is ...

This is not true in general. Take your example and reembed everything using the $n$-th Veronese map. A subvariety $F$ of $X$ is reembedded as a variety of degree ${\rm deg}(F) n^{\dim(F)}$. In ...

Any line on $X$ has at least one point in common with the plane $x_2=0$. Thus, any line on $X$ contains a point whose first two coordinates vanish. In particular, one of the equations of a line in $...

Maybe I am making a mistake, but you could try the following. First, for each $k$ the set $U_k$ of points $x$ for which there is a section of $L^k$ not vanishing at $x$ is open. Moreover, the union ...