dhy
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IMO 2017/6 via arithmetic geometry
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33 votes

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}}\...

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The most outrageous (or ridiculous) conjectures in mathematics
33 votes

The "conjecture" in algebraic geometry that all rationally connected varieties are unirational comes to mind. It's usually thrown around as a way of saying, "See, we know so little about what ...

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Which hypersurfaces in $\mathbb{P}^n$ are abelian varieties?
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21 votes

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 ...

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Short exact sequences every mathematician should know
20 votes

The Tate extension. Let $k$ be a field, and let $V$ be the space $k((t))$ be the space of Laurent series with coefficients in $k$, considered as a topological vector space. If we write $\operatorname{...

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Regarding the irreducibility of certain varieties
13 votes

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^...

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Alice and Bob playing on a circle
13 votes

Here is an argument that Alice wins for odd $n$. Label the points $1$ to $n.$ Lemma: If at any point Alice has marked $k$ points in a row and no other points, Bob has marked the two points ...

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What exact number of domino tilings cannot be realizable?
13 votes

Consider a $4\times 2k+1$ rectangle. Label its points by $(x,y)$, with $x$ from $0$ to $3$ and $y$ from $0$ to $2k.$ Remove from the rectangle the points $(1,y)$ and $(2,y)$ for $y$ odd. I claim that ...

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What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?
11 votes

An update from 2018: There has been some recent work on a heuristic suggesting that there are infinitely many elliptic curves of every rank $<21$ but only finitely many of rank $> 21$ (it's ...

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Why do we care about $(\infty,2)$-categories?
10 votes

One place where $(\infty,2)$-categories shows up is the geometric Langlands program. (As in David Ben-Zvi's comment, this is again related to the TFT example.) Indeed, local geometric Langlands is ...

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How to see that the determinant of this matrix is nonzero for all primes?
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10 votes

As discussed in the comments, I don't see how to extract the desired matrix from the original question (about spanning the vector space). However, the matrix being nonzero IS equivalent to the ...

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Simple question about polynomials
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9 votes

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 ...

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Ideal of the boundary of $G/U \subset \overline{G/U}$
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8 votes

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 ...

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What are advantages of chiral algebras over vertex algebras?
8 votes

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. ...

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Remark 2.4.1.4 Higher Topos Theory
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7 votes

Let's see what the data of a diagram \begin{matrix} \partial\Delta^{n-2}&{\to}&X_{/f}\\ \downarrow &&\downarrow\\ \Delta^{n-2}&{\to}&X_{/y}\times_{S_{/p(y)}}S_{p(f)}\end{...

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How should I think about the Grothendieck-Springer alteration?
7 votes

The answer to 1) is that this is a special case of a broader phenomenon for symplectic resolutions (though I think some features are specific to the Grothendieck-Springer case.) For instance you have ...

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Examples of conjectures that were widely believed to be true but later proved false
7 votes

Lusztig's conjecture in modular representation theory, which would describe the characters of simple $G_1$-modules for $p$ larger than the Coxeter number, was generally believed to be true for a long ...

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If I exchange infinitely many digits of $\pi$ and $e$, are the two resulting numbers transcendental?
7 votes

Here's an argument that one should expect that the two numbers gotten this way must be transcendental. Really, what I am showing is that the locus of $(a,b)$ in $\mathbb{R}\times\mathbb{R}$ where one ...

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Application of toric varieties for problems that do not mention them
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7 votes

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 ...

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Relation between flatness and integrability of an algebraic connection
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6 votes

(In characteristic zero) Flatness implies the other two definitions; integrability and formal lifting are very weak conditions (in fact if I haven't made a mistake I think this notion of integrability ...

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Conceptual explanation for curious linear-algebra fact in characteristic $2$
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6 votes

I think one way of explaining this is via quadratic forms. The usual correspondence sends $X$ to the quadratic form $X(v)=vXv^t,$ $v$ a row vector. But in characteristic $2$, this formula simplifies ...

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Why should affine lie algebras and quantum groups have equivalent representation theories?
5 votes

(Written on my phone - apologies for any typos.) A few comments: a) First, as to the source of the braided monoidal structure on the Kazhdan-Lusztig category. The category of integrable affine Lie ...

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Thomason-Trobaugh Theorem
5 votes

All perfect complexes on an open $U$ of a smooth variety $X$ extend to perfect complexes on $X$. In fact, I believe a much more general statement is true: For an open substack $U$ of a separated ...

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Universal enveloping algebra and the algebra of invariant differential operators
5 votes

From the Lie algebra inclusion $\mathfrak{g}\rightarrow \Gamma(T_G)$ of right invariant vector fields we get a map $U(\mathfrak{g})\rightarrow\Gamma(\mathcal{D}_G).$ This induces a map $U(\mathfrak{g})...

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$K_0$-equivalence of varieties
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5 votes

(Expanding my comments into an answer for more visibility.) By Larsen-Lunts $K_0(\operatorname{Var}_k)/[\mathbb{A}^1]$ is the free abelian group on stable birational equivalence classes. It thus ...

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Holomorphic vector bundles over a Riemann surface does not satisfy $\mathbf{AB2}$ but satisfies $\mathbf{AB1}$
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5 votes

(This should really be a comment on S. Carnahan's answer, but don't have enough reputation.) All coherent sheaves on a Riemann surface split into the direct sum of a torsion sheaf and a vector bundle....

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A basis of holomorphic differentials on Fermat curves
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4 votes

The automorphisms $a_1,a_2,a_3$ are part of a $\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$ action on $F_k$, which in turn induces an action on the space $H^0(\Omega^1_{F_k})$ of holomorphic ...

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Drinfeld Sokolov and the semiinfinite flag variety
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4 votes

Maybe let me try to synthesize my comments into an answer. All of this is contained in Raskin's beautiful paper arxiv.org/abs/1611.04937 on Whittaker categories. Convention: We work here in the ...

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Real plane cubic curves from points in Gr(3,6) via a certain 6x6 determinant
4 votes

Your map (assuming that the determinant is nonzero for all elements of the grassmannian) is a projection of the Plucker embedding. This tells you that the degree of the corresponding map of complex ...

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Secant varieties of curves in $\mathbb{P}^4$
4 votes

I think the answer is no. Here is an argument. Degenerate your $8$ points so that there are $2$ $5$-tuples that lie in $3$-planes $A,B$. The degree $3$ rational curves through the first $5$-tuple ...

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Do the following two filtrations of the affine Grassmannian agree?
Accepted answer
4 votes

This statement isn't true; as you've currently defined everything, your $X_i$ are infinite dimensional (in fact they are unions of connected components.) You want instead to take the $X_i$ unions of $\...

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