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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classifying closed subschemes of an affine scheme

Looking at the Stacks Project proof that every closed subscheme of an affine scheme $\operatorname{Spec} A$ is of the form $\operatorname{Spec} A / I$, where $I$ is an ideal of $A$. It uses the fact ...
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Adjunction formula in Griffiths Harris

In the book Griffiths and Harris, they states that: $N^*_V=[-V]|_V$, here $V$ a smooth hypersurface, $N^*_V$ means the conormal bundle w.r.t. $V$. They define the transition function of $V$ as $g_{\...
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Is prime spectrum the soberification of maximal spectrum?

Given a commutative ring $R$, let $\operatorname{Spec} R$ denote its set of prime ideals, equipped with the Zariski topology. Let $\operatorname{Spm} R$ denote the maximal spectrum of $R$, which is ...
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Bertini's Theorem induces holomorphic mapping

I'm learning Bertini's Theorem via the famous book by Griffiths and Harris. In the page 138 they say that, if the linear system(over manifold $M$) we consider is just a pencil $\{D_{\lambda}\}, \...
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Some questions about the closure of the graph of projection $\Bbb P^n\dashrightarrow\Bbb P^{n-1}$ from Mumford's book.

The following is from David Mumford's Algebraic geometry I Complex Projective Varieties, page 32. Let $\mathbb{P}^n_X$ and $\mathbb{P}^{n-1}_Y$ be spaces with coordinates $(X_0,...,X_n),(Y_0,...,Y_{n-...
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Determine the ideal of the algebraic set $V(X^2Y, (X-1)(Y+1)^2)$

I want to determine the ideal of $V(X^2Y, (X-1)(Y+1)^2)$. I know algebraic geometry only at a very beginner level so I'm sorry in advance if my argument is trivially flawed. This is what I've done: ...
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Combinatorial description of hard Lefschetz for toric manifolds

Let $X$ be a toric variety which is compact and smooth (feel free to correct me if I’m missing adjectives), and let $\Sigma$ be the corresponding fan. The ordinary cohomology ring $H^*(X)$ has a ...
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Show $y^4 - x^5$ is irreducible in $\mathbb C[[x,y]]$ [duplicate]

Is there a 'conceptual' way to see that $f(x,y) = y^4 - x^5$ is not the product of two power series $a(x,y)$ and $b(x,y)$ unless either $a$ or $b$ are invertible? I guess I am thinking of $\mathbb{C}[[...
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Exceptional divisor of blow-up in a non-rational point

Let $X$ be an prjective variety over $k$ and $Spec(L) \to X$ be a point of degree $n$ on $X$. Is there any description of $Bl_{Spec(L)}X$ in terms of $L$? Is there any connection between them in the ...
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Differential form in a variety, from definition and properties

Following Shafarevich Reid - Basic algebraic geometry 1, the differential at $p$ of a polynomial $F(T_1,\dots,T_n)$ is the linear part of the Taylor expansion at $x$ (p. 87), so $$ (dF=)d_xF=\sum \...
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On global section of a line bundle.

At Proposition 1.1.1. (1) of https://arxiv.org/pdf/0706.0494.pdf it is written that $H^0(X, L \otimes {\cal{J}}(X, ||L||)) = H^0(X, L)$ ensures that every global holomorphic section $s$ of $L$, i.e., $...
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Existence of finitely many global secitions which generate $\mathcal{O}_X$-module of finite type

I have a question. Let $X$ be a quasi-compact scheme. Let $\mathcal{F}$ be a $\mathcal{O}_X$-module of finite type (locally generated by finitely many sections). Assume that $\mathcal{F}$ is genetaed ...
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For Divisors, do we use order of the point or multiplicity of the point?

This is from the Book Mathematical Cryptography by Silverman, Pipher etc. From the book where they describe Rational Functions & their Divisors on Elliptic Curves 5.8.2 Rational functions and ...
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$M$ is a real analytic variety and $p \in M$ such that $M \setminus \{p\}$ is a manifold so $M$ is a real analytic variety in a neighbourhood of $p$

Suppose that $M$ is a real analytic variety and $p \in M$ such that $M \setminus \{p\}$ is a $3-$dimensional (real) manifold. Therefore, $M$ is a real analytic variety in a neighbourhood of the point $...
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Basis of holomorphic 1-forms on smooth projective curve with affine equation $C\colon s^6 = t^{p+1}+1$

Let $k = \mathbb{F}_{p^2}$ be a finite field with $p \equiv -1 \mod 6 $. Via Riemann-Hurwitz one can calculate that the genus of this curve equals $5(p-1)/2$ and hence the the space of holomorphic ...
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Some notation in definition of smooth morphism between schemes.

I'm reading Gortz's Algebraic Geometry, Defitnion 6.14 and I'm confused about some notation : What ${\partial {f_i}\over \partial {T_j}} (x)$ does exactly means? As he remarked below, for $g = {\...
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For all $ m$ and $ n,$ there exists an $n \times m$ matrix $ A$ such that $ A^TA=\frac{n}{m}\,I$ and every element of the diagonal of $ AA^T$ is $ 1$?

The following speculation comes from this thread, which was inspired from a Putnam competition problem. For all $ m$ and $ n,$ there exists an $n \times m$ matrix $ A$ such that $ A^TA=\frac{n}{m}\,...
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Interior of the image of a morphism

Let $\phi:X\to Y$ be a morphism between irreducible quasiprojective varieties. If $\phi$ has a dense image in $Y$ can we conclude that its image has an interior? It really feels like it, but I couldn'...
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Is a product of curves a complete intersection?

Let $C_1, C_2$ be two projective smooth curves over $\mathbb{C}$. Is it possible to say when $C_1 \times C_2$ a complete intersection in some projective space? For three curves the answer is "it ...
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Birational Equivalence

On p.$12$ of Shafarevich's "Basic Algebraic Geometry I", a rational map, say $\phi$, from the irreducible algebraic plane curve $X: f(x,y) = 0$ into the curve $Y : s^2 - p = 0$ is given by (...
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Is numerical equivalence preserved by base change?

Vakil's Foundations of Algebraic Geometry, Proposition 20.1.4 is that intersection multiplicity depends only on numerical equivalence classes. The proof uses base change to an algebraically closed ...
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canonical bundle of a curve on a surface

Let $C$ be a nonsingular curve of genus $g$ on a surface $X$, then we know by Harthsorne, Chapter-$2$, Proposition $8.20$ that $K_C \cong K_X \otimes \mathcal O_X(C) \otimes \mathcal O_C$, then can ...
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normal noetherian schemes admit a connected affine open cover

I'm recently working on normal schemes and locally factorial schemes. The following is a result I need, which I believe is true, but I don't know how to prove it. Let $X$ be a noetherian normal scheme....
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Does a torsion-free coherent sheaf embed into a locally free sheaf?

Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...
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How does one construct an affine variety morphism from an algebra homomorphism between their coordinate rings?

$K$ denotes an algebraically closed field. My course states that the category of affine varieties is equivalent to the opposite if the category of finitely generated reduced $K$-algebras. If $V \...
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Silverman Proposition 2.5 computation

In the proof of Proposition 2.5 in Silverman's Arithmetic of Elliptic Curves, the author defines a map $$E_{ns} \to \overline{K}^*, \quad [X,Y,Z] \mapsto 1 + \frac{AX}{Y},$$ where $E_{ns}$ is the ...
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How to show $\operatorname{Pic}(X)=0$? Exercise $14.2$.Q Vakil's notes

I'm reading Vakil's notes and I'm struggling with the exercise $14.2$.Q. I've been able to prove everything except $\operatorname{Pic}(X)=0$ with $$ X=\operatorname{Spec}\frac{k[x,y,z]}{(xy-z^2)}. $$ ...
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different torsor structures on the same scheme?

Exercise 10.4 in Silverman's Arithmetic of Elliptic curves states that two torsors $X/K$, $Y/K$ for the same elliptic curve $E/K$ are isomorphic as $K$-varieties if and only if the classes $[X]$, $[Y]$...
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$\operatorname{Supp}(\mathcal{O}_X/\mathcal{I}) \subseteq \operatorname{Supp Coker}(u_f)$?

Let $X$ be a quasi-compact, quasi-separated scheme and let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\mathcal{I} $ be a quasi-coherent ideal of $\mathcal{O}_X$ of finite type (...
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Extension by a skyscraper sheaf

Let $C$ be a nodal curve. Then we have the normalisation map $f: \mathbb{P}^1 \to C$. I want to understand the sheaf $f_* \mathcal{O}_{\mathbb{P}^1}$. There is the map $\mathcal{O}_{C} \to f_* \...
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The local ring at the origin of $ \Bbb{A}^2 $

Consider the curves $ F = y-x^3 $ and $ G = y^3-x^4 $ over $ K. $ Find a polynomial representative of $ \frac{1}{1+x} $ in $ \mathscr{O}_0/ \langle F,G \rangle. $ I am having trouble simplifying $ \...
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slope and morphism of bundles

Let $L_1, L_2$ be two line bundles on a smooth projective variety $X$. Consider that there exists a non-zero homomorphism $f:L_1 \to L_2$. Over curve this gives us $\text{deg}(L_1) \leq \text{deg}(L_2)...
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Definition of restriction of morphism of ringed space?

I have a question. Is there a definition of restriction of morphism of (locally) ringed space? Let $(f,f^{\flat}): X \to Y$ bea morphism of ringed spaces ; i.e., $f:X\to Y$ is a continuous map and $f^{...
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Discrepancy between order of the identity element & its inclusion in n-torsion sets

The order of the identity element of a group is usually considered to be one because $1 * e = e$ or $e^1 = e$. However, the text on torsion points (from Mathematical Cryptography by Silverman, Pipher ...
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Find the radical of an ideal in the ring of polynomials over the complex numbers

In $\mathbb{C}[X,Y,Z,W]$ I have the following ideal $$I=(XZ-Y^2,XW-YZ,X^2-WY,YX-WZ,YW-Z^2,ZX-W^2).$$ I'm trying to find it's radical using Hilbert Nullstellensatz, i.e., trying to find the set of ...
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What is a generically reduced scheme?

I am reading the book "3264 & All That Intersection Theory in Algebraic Geometry". In the following definition (see page 30) Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of ...
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Fibers of the Abel Jacobi map over curves

I am studying the Abel Jacobi map $$\mathrm{Div}_{X/k} \to \mathrm{Pic}_{X/k}$$ for projective, smooth, irreducible curve $X/k$ where $k$ is algebraically closed. Let $S = \operatorname{Spec}(k)$, $T$ ...
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Non singular point of a projective plane curve

In problem 5.1 of Fulton's Algebraic curves, we're asked to show that a point $P\in\mathbb P^2$, $P=[P_1:P_2:P_3]$ is multiple iff $F(P)=F_X(P)=F_Y(P)=F_Z(P)=0$. Here $P$ is said to be multiple if $...
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In the proof of characterization of ampleness of $\mathcal{O}_X$-module

I'm reading the Gortz, Algebraic Geometry and now trying to understand some statement. First, let me arrange some related definitions. Let $X$ be a quasi-compact and quasi-separated scheme. An ...
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How to calculate the pullback of sheafs

I am trying to look at a concrete example of the pullback of sheaves. The easiest example to consider might be the closed embedding. $i: \operatorname{Spec} k[x] / (x) \to \operatorname{Spec} k[x]$. ...
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transcendental dimension of a variety

I am trying to understand the definition of the dimension of a variety using the notion of a transcendental basis. Consider for an algebraically closed field $\mathbb{K}$ the variety $V=\left\{(x,y)\...
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Descent of sheaves to an Enriques surface with endomorphism algebra isomorphic to a matrix algebra

Let $X$ be an Enriques surface with covering K3 surface $p: Y\rightarrow X$ and associated involution $\iota$. If $E$ is a simple coherent sheaf on $Y$, i.e. $\mathrm{End}_Y(E)\cong \mathbb{C}$, such ...
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Intuition between the equivalence between Cech and Singular Cohomology?

We know that, under suitable assumptions, the Cech Cohomology of a topological space is isomorphic to the singular cohomology. The proof seems to be mostly algebraic. I am wondering: is there a ...
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Adele space of rational function field

Let $F=K(x)$ and $K(x)/K$ be the rational function field. I am then trying to prove \begin{align*} A_F = A_F(0) + F \end{align*} where $A_F$ is the adele space, $A_F(0)$ the set of adeles $\alpha$ ...
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Geometric Interpretation of Jacobson rings -- Every locally closed subsets of $\operatorname{Spec} A$ consists of a single point is closed

I'm currently working on Atiyah&MacDonald's book on commutative algebra. I'm trying on Exercise 5.26, and remaining to show that Question: $A$ is a Jacobson ring if and only if $$ \text{every ...
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definition of closed immersions of schemes

Qing Liu's algebraic geometry and arithmetic curves defines closed immersion as follows(definition 2.22): We say that a morphism $(f, f^{\#}): (X, O_{X}) \to (Y, O_{Y})$ is an open immersion (resp. ...
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Pullback is not exact

Let $f: X \to Y$ be a map of schemes. Then we have the "quasicoherent" pullback which takes $F \in Qcoh(Y)$ and gives $f^* F = \mathcal{O}_{X} \otimes_{f^{-1} \mathcal{O}_{Y}} F$. This ...
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Decomposing a big and nef divisor into ample + effective

Let $\pi: X \to \mathbf{P}^2$ be the blow-up of the projective plane at one point. Write $H$ for a hyperplane divisor of $\mathbf{P}^2$. The pullback $\pi^* H$ is big and nef, so it can be written in ...
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What is the right adjoint to the functor $\sf{Psh}\to\sf{Set}$ which evaluates the presheaf on the whole space?

$\newcommand{\O}{\mathcal{O}}\newcommand{\T}{\mathcal{T}}\newcommand{\op}{^{\sf{op}}}\newcommand{\set}{\sf{Set}}\newcommand{\ps}{\sf{Psh}_{\T}}$Let $\T$ be a topological space and $\O(\T)$ the poset ...
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What are the limits on the denominators of rational points of varieties?

I have two large polynomial (multinomial) equations in two variables with integer coefficients. How can I calculate the largest denominator of any simultaneous rational solution?
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