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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

1 vote

Why do we make such big deal about the 'unsolvability' of the quintic?

I like this question because I agree with its sentiment. Let me give an additional reason why the insolubility of the quintic is an overrated result in my opinion. I believe that we shouldn't even be …
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2 votes

Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$

This also follows from Prop. 2.3.26(i) in Bjorn Poonen's Rational Points on Varieties, where it is stated that if for a finite type $k$-scheme $X$ the set of rational points $X(k)$ is Zariski dense, t …
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3 votes

Generalization of Weak Nullstellensatz?

Or see Proposition 2.4.6 in Bjorn Poonen's book Rational Points on Varieties (link). This is almost exactly the result you conjectured, just a bit more general: Let $X$ be a $k$-variety. Then the map …
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6 votes

Conic sections are to cones as quadric surfaces are to what?

The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would sa …
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3 votes
Accepted

What is the state-of-the-art for solving polynomials systems over fields that are not algebr...

For the reals, I particularly like the book by Sturmfels mentioned by Alexandre Eremenko. For the rational numbers, you can hardly do better than Bjorn Poonen's book Rational Points on Varieties, whic …
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5 votes
2 answers
1k views

Singular models of K3 surfaces

Let us work over a ground field of characteristic zero. As is well-known, a K3 surface is a smooth projective geometrically integral surface $X$ whose canonical class $\omega_X$ is trivial and for whi …
5 votes

Singular models of K3 surfaces

For what it's worth, I wrote up a proof (pretty detailed) following the hints in Francesco Polizzi's answer. It's in an unpublished preprint found here (p. 38 onwards). I am not a geometer, so the exp …
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19 votes

Results that are widely accepted but no proof has appeared

I think one example is given in this MO question of mine: a quartic in $\mathbb{P}^3$ with at worst Du Val singularities is a K3 surface (and similar statements for two types of complete intersections …
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6 votes

Two queries on triangles, the sides of which have rational lengths

If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$ pqr(p+q+r) = A^2. $$ Now I knew I recognize …
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4 votes

A curve is proper iff the space of global sections is finite-dimensional

The answer to the first question is certainly yes, because if a curve is non-proper, it must be affine, and hence its ring of global sections is not finite as a $k$-module. See this link to a M.SE top …
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8 votes
Accepted

Rational points on open subsets of affine space

Here is a short proof that, for an infinite field $k$, and all non-zero polynomials $F \in k[x_1,\ldots,x_n]$ in $n$ variables, there exists an $n$-tuple $a_1,\ldots,a_n \in k$ such that $$ F(a_1,\ldo …
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16 votes
Accepted

reference request: rational points on the unit sphere

The earliest reference is surely Diophantus' Arithmetica. His "method of adequality" can be used to construct rational points on quadrics that approximate real points arbitrarily well (that is, starti …
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7 votes

Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

The proof given in Otto Forster, Lectures on Riemann Surfaces (Graduate Texts in Mathematics 81), chapter 16, seems very much suited to your list of prerequisites.
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12 votes
0 answers
265 views

Galois groups of classical differential equations

I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following: Is there a book or article devoted (either partially or completely) to …
10 votes
Accepted

Does GAGA hold over other topological fields?

If $k$ is a field that is complete with respect to some ultrametric valuation, then there is the "GAGR" (i.e. géométrie algébrique et géométrie rigide) theorem. A succinct explanation (in French, with …
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