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The set $S$ gives rise to a subscheme (which let's also denote by $S$) of $\mathbb{P}^1_{\mathbb{Z}},$ because relatively a prime pair $(x,y)$ corresponds to a section of $\mathbb{P}^1_{\mathbb{Z}}\ri …

The Beilinson-Bernstein localization theorem states roughly that the category of $D$-modules on the flag variety $G/B$ is equivalent to the category of modules over the universal enveloping algebra $U …

Really this is mostly just consolidating what has been (implicitly) said in the comments and cleaning it up a bit (e.g. using the Chow ring instead of singular cohomology), but might as well make it a …

Let's say I have a smooth irreducible subvariety $X$ of $\mathbb{CP}^n$ with some fixed Hilbert polynomial. What are the best bounds known for the sum of the Betti numbers of $X$? That such a bound ex …

No, take $f=b^4-a^3c$ and $g=ac^3.$ Now the variety $V(b^4-a^3c,y^2-ac^3)$ being reducible is equivalent to the ideal $(b^4-a^3c,y^2-ac^3)$ being prime. But we have $(ya-b^2c)(ya+b^2c)=y^2a^2-b^4c^2=a …

Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot o …

A cubic threefold is a smooth degree $3$ hypersurface in $\mathbb{P}^4$. Is there a cubic threefold $X$ over any field $k$ (possibly of positive characteristic) and an odd degree rational map $\mathbb …

There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not Z …

As pointed out in the comments, this is Theorem 6.3.1.2 of Hilburn-Raskin. (It certainly was known much earlier, but I'm not sure what to give as a reference.) Their proof is stated quite elegantly, i …

Note that the defining equation for $X$ can be rewritten as $$(x+y)^3+(x-y)^3+(z+w)^3+(z-w)^3=-2.$$ As the linear transformation $(x,y,z,w)\mapsto(x+y,x-y,z+w,z-w)$ is invertible over any field of cha …

This equation has no solutions when $d$ is odd. (EDIT: See below for the general case.) Actually, for $d$ odd, there are no triples $(F_0,F_1,F_2)$ with $F_1\cdot F_2-F_0^2$ a multiple of $x$, let al …

Some comments: It is not necessarily true that chiral algebras are essentially conformal vertex algebras, as chiral algebras are allowed to vary over the curve in a way that vertex algebras are not. …

Here is one way to see it, via classifying $G$-invariant radical ideals. (This has the bonus that it implicitly describes the boundary.) Lemma: $G$-invariant ideals $I$ of $\mathbb{C}[G/U]$ are in bij …

There are lots of applications of toric varieties to singularities, e.g., the proof of the weak factorization theorem in characteristic zero. (Indeed, the name of the linked paper is "Torification and …

In section 10.4 of "Vertex Algebras and Algebraic Curves", Ben-Zvi & Frenkel (second edition), the authors claim that for any vertex algebra V, the space of one-pointed conformal blocks with insertion …