Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

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Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this (Proposition 2.2.20, p. 97): Let $\mathbb{k}$ be an ...
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Recent developments in the proof of fermat's last theorem

It's been 20 years since fermat's last theorem was proved by Andrew Wiles. Has there been any simplification in proof in the last 20 years? What I do only know is that different proofs of faltings's ...
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When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that $f$...
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Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
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When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
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When is the direct image functor exact?

Consider a morphism of topological spaces $f:X\to Y$. The direct image functor takes a sheaf $\mathcal{F}$ on $X$ to the sheaf defined by $f_*\mathcal{F}(U)=\mathcal{F}(f^{-1}(U))$. It's a right ...
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Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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(Weil divisors : Cartier divisors) = (p-Cycles : ? )

Suppose $X$ verifies the suitable conditions in which Weil (resp. Cartier) divisors make sense. The group of Weil divisors $\mathrm{Div}(X)$ on a scheme $X$ is the free abelian group generated by ...
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Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
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Why is the sheaf $\mathcal{O}_X(n)$ called the "twisting sheaf" (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
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The Hasse invariant at cuspidal elliptic curves in a generically ordinary family

For an elliptic curve of the form $y^2 = f(x)$ where $f(x) \in \mathbb F_q[x]$ is a cubic polynomial with distinct roots, it is known (from Silverman's book, say) that the curve is supersingular ...
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Can every variety appear as singular locus?

let $V$ be a projective variety. Does there always exist another variety $X$ such that $\operatorname{Sing}X=V$? Here $\operatorname{Sing}X$ means the singular locus of $X$ with reduced structure.
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Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are some of ...
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Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
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Transformations that map points inside the sphere to points inside the sphere

I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by ...
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Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
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In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to \$\textrm{Spec } S_{(...