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 $\mathbb{... View answer 7 votes Just to complement quim's answer, note that if$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$is ... View answer Accepted answer 7 votes Let$-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$and the restriction of the intersection pairing to the ... View answer 6 votes Assume that$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$(e.g. restrict ... View answer Accepted answer 5 votes Asking individual vertices, you figure out the valence of each vertex with$n$questions. Asking for pairs of vertices, you can then decide if each pair has an edge between them or not: they have an ... View answer Accepted answer 4 votes Per Martin's request, here is a more detailed version of my comment. Given a graph$G$, the independence game is the game in which two players take turn in removing a vertex and all of its neighbors. ... View answer 4 votes I do not know a reference, but the following short argument seems to work. Assume that the dimension of$X$is at least 1! Argue by induction on the dimension of$X$. Reduce to the case in which ... View answer 2 votes I am not too familiar with algebraic spaces, but maybe this is an example of an algebraic space that cannot be a quotient of a scheme by a finite group. The property of such quotients that I use is ... View answer 2 votes This is not true in general. Take your example and reembed everything using the$n$-th Veronese map. A subvariety$F$of$X$is reembedded as a variety of degree${\rm deg}(F) n^{\dim(F)}$. In ... View answer 2 votes Any line on$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 ...