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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.

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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Canonical log structure defined by a normal crossing divisor

Given a locally noetherian scheme $X$ and $D \subset X$ a normal crossing divisor. Let $j: U=X-D \hookrightarrow X$ the open complement immersion. Define the log structure $(M_X,\alpha_X)$ on $X$ as the direct image by $j$ of the trivial log…
user48900
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Holomorphic differential on $y^3=x^5-1$

I'm trying to study the canonical map $\phi_K$ for the algebraic curve $\mathcal{C}:y^3=x^5-1$ and to do this I need to find a basis for $\Omega^1(\tilde{\mathcal{C}})$ where $\tilde{\mathcal{C}}$ is the nonsingular model (the curve is singular at…
cartesio
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Why is a theorem about extending morphisms $\operatorname{Spec} K\to \mathbb{P}^n_K$ called "The Heavenly L'Hopitals Rule"?

In Introduction to Schemes, G. Ellingsrud and J. C. Ottem call the result below "The Heavenly L'Hopitals Rule". I see absolutely no similarity between this and the usual L'Hopital rule. What do they mean by this name?
Gabriel
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Quotient category $Coh_d/Coh_{d-1}$

Let $X$ be a scheme, and $Coh(X)_d$ be the category of coherent sheaves of support <= d dimensional. Why is $Coh(X)_d/Coh(X)_{d-1}$ equivalent to $\oplus_{x \in X^{(d)}} \mathcal{A}(\mathcal{O}_{X,x})$? See:…
user5262
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Fulton Algebraic Curves: Exercise 3.12

I am trying to understand for which $n$ does the curve $F = Y - X^n$ has an inflection point at $P = (0,0)$ as in exercise 3.12 from Fulton's algebraic curves. From a geometric perspective I expect it to be all $n \ge 3$ with $n$ being odd. The…
G.rald
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The Inverse of a Rational Section is a Rational Section of the Dual

This question comes after reading the last paragraph of Vakil's FOAG, p. 400. We consider the set of $\{(\mathcal L, s) \}$, where $\mathcal L$ is an invertible sheaf on a Noetherian, reduced, regular in codimension 1 (in case any of that matters)…
Johnny Apple
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Geometric Nakayama's Lemma

This is Vakil 13.7 E, self-study. We are to show that if $X$ is a scheme and $\mathcal F$ is a finite type quasicoherent sheaf on $X$, then if $p \in U \subset X$ is an open neighborhood of $p$ and $a_1, ... , a_n \in \mathcal F(U)$ have images…
Johnny Apple
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Vanishing criterion for sections of module on a product

This queston can be regarded as a variant of this one: Does a section that vanishes at every point vanish? Let $X,Y$ be varieties, $M$ be an $\mathcal{O}_X$-Module and $$\pi:X\times Y\rightarrow X$$ be the projection. Let $s$ be a local section of…
Jan
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Does $p(a) = p(b) \Rightarrow a=b \ $?

Let $S = (P_1,P_2, \ldots,P_n)$ be a set of polynomials with complex coefficients. I call S critical if set of solutions of $\{P_n(a) = P_n(b)\forall n \}$ in $\Bbb C^2$ is $\{ (a=b) \cup (\text{some finite points in }\Bbb C^2)\}$ I call a set…
rohit
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How to construct line bundles of degree $g-1$ on smooth projective curve with no global section?

Let $C$ be a smooth projective curve of genus $g$, we know that for a general line bundle $\mathcal{L}$ of degree $g-1$, $\mathcal{L}$ has no global sections, i.e. $\text{H}^0(C, \mathcal{L})=0$. My question is that, in some particular case, for…
Xueqing Wen
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Arguments on translation of affine space.

Let $k$ be an algebraically closed field. I have found in some algebraic geometry books arguments such as "by translation we may suppose that every maximal ideal $\mathcal{m} = \langle x_1-a_1, \ldots, x_n-a_n \rangle$ is of the form $\mathcal{m} =…
HeMan
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A finite type morphism between regular schemes and closed immersions.

I am working through the following problem in Qing Liu's book on Algebraic Geometry, 6.2.6 a), which reads: Let $X \rightarrow Y$ be a morphism of finite type of locally noetherian regular schemes. Let $y \in Y$ and let $x \in X_y$ be a closed…
Dedalus
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What does the $f^!$ do for line bundles on a curve?

I'm trying to understand what does inverse exceptional image $f^!$ of a coherent sheaf look like. Let's say for the sake of this question that $f:X\rightarrow Y$ is a finite flat morphism between schemes. In this case, my understanding is that $f^!$…
xlord
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Defining localization of modules through universal property

On page 33 of Vakil's book on Algebraic Geometry, he shows how one can define the localization of modules purely in terms of universal property and later shows that a specific definition satisfies the property. Basically he says that if $M$ is an…
Akash Kulkarni
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A principal open set of an affine algebraic set is an affine variety

Notations $k$ is an algebraic closed field and $\mathbb A^n(k)$ is the topological space $k^n$ with the Zariski topology If $X\subseteq\mathbb A^n(k)$ is an affine algebraic set and $f\in\Gamma(X)$, then $D(f)=\{x\in X\,:\, f(x)\neq0\}$ An affine…
Dubious
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