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. ...

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How to prove this comment of Fulton

I'm trying to understand why this is true in Fulton's Algebraic Curves: Why we add this point $(0,\ldots, 0)$? Why this equality is true? I really need help. Thanks in advance.
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Thomason resolution of sheaves

Let $X$ be a smooth quasi-projective scheme over a field $k$ and $G$ an algebraic group (also over $k$ not necessarily reductive) acting on $X$. I the work of Thomason "Equivariant Resolution, ...
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Finite surjective morphism of smooth varieties is flat

Let $f: X \to Y$ be a finite surjective morphism of nonsingular varieties over a field $k$. Exercise III 9.3. in Hartshorne's Algebraic Geometry sais that if $k$ is algebraically closed, then $f$ is ...
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Finding an algebraic equation given divisors

I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this: Given divisors $(\omega_1) = P_1 + 5 P_2 + 2 P_3,$ $(\omega_2) = 5 P_1 + P_2 + 2 P_3,$ ...
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32 views

Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...
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48 views

Closed immersion factors through closed immersion

I'm currently working through the proof of Theorem 1, III.12 in Mumford's "Abelian Varieties". Let $G$ be a finite $k$-group scheme acting on an affine $k$-scheme $X:=\text{Spec}~A$ and let $\phi: ...
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41 views

Hartshorne P24 Lemma 4.1

Lemma 4.1 says:let $X$ and $Y$ be two varieties, and let $\phi$ and $\psi$ be two morphisms from $X$ to $Y$, and suppose there is a nonempty open subset $U \subseteq X$ such that $\phi|_U = \psi|_U$, ...
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79 views

Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $

Let $\mathbb{P}^1 $ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \ $ we can define some quasi-coherent sheaves on $\mathbb{P}^1 $. We can ...
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47 views

Definition of Hodge structure: is torsion allowed?

I am trying to understand the definition of an integral Hodge structure. Apparently, for $X$ a compact Kahler manifold, $H^n(X,\mathbb R)$, the lattice $H^n(X,\mathbb Z)$ and the Hodge filtration give ...
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11 views

Find the operators corresponding to partial derivatives, under a given polynomial automorphism

Suppose $$F=(F_1,\ldots,F_n) \colon \mathbb{C}^n \to \mathbb{C}^n$$ is a polynomial automorphism (i.e. invertible, and the inverse is also a polynomial map). (I'll use $\mathbb{C}[X]$ as a shorthand ...
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26 views

Why are morphisms of schemes locally given by ring homomorphisms?

I am trying to prove the statement: A morphism $f:Y \rightarrow X$ of schemes is determined locally by homomorphisms of rings. Here is my attempt: Given the morphism $f:Y \rightarrow X$ and any ...
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37 views

Field of Definition of an Algebraic Group

Linear Algebraic Groups- James E. Humphreys Chapter-XII Let $K$ be an algebraically closed field and $k$ be a arbitrary sub-field of $K.$ A closed set X in $A^n=K\times ...$(n times)$\times K$ is ...
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24 views

A have a quick question about moishezon manifold

Is moishezon manifold general type?
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42 views

About fibers of a morphism

Suppose $f:X\to Y$ is a surjective morphism bewteen algebraic varieties , does the locus of non-reduced fibers form a closed subset of $Y$? If the condition is not good enough, one may add ...
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54 views

Calculating a dual variety to a Chow variety

I am unfamiliar with algebraic geometry, yet I am faced with calculating three special cases of the following (the full text can be found at http://arxiv.org/abs/1107.4659) The $n^{\times ...
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1answer
24 views

Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
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1answer
36 views

Examples of varieties with torsion in their integral Hodge structure

I am not so used to thinking about integral Hodge structures, so this question might be completely trivial. What are easy and interesting examples of smooth projective connected varieties $X$ with ...
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141 views

Surjective morphism from $X$ to itself is finite

Let $X$ be a projective variety. I would like to prove that any surjective morphism $f: X \rightarrow X$ is finite, but I can't see a good strategy. Any hints?
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66 views

Bijection between solutions of polynomials equations and spectrum of the quotient ring.

I would like to see the connection between the set of solutions of a system of polynomial equations and a spectrum of the quotient polynomial ring. Given $f_1, f_2,\cdots, f_m \in k[x_1,x_2,\cdots, ...
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55 views

Two questions on the definition of $\mathcal{O}_X(U)$ for an affine scheme $X$.

Let $X=\operatorname{Spec}(A)$ be an affine scheme. Hartshorne defines $$ \mathcal{O}_X(U)=\{s\colon U\to\coprod_{\mathfrak{p}\in U} A_\mathfrak{p} \mid s(\mathfrak{p})\in A_\mathfrak{p} \text{ and } ...
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39 views

Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
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35 views

Where can I find some articles of Weil.

Where can I find the articles of Weil: Variétés abéliennes et courbes algébriques Sur les courbes algébriques et les variétés qui s'en déduisent. on Internet?
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Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
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41 views

Can a rational map $X\leadsto Y$ be defined as a scheme morphism $Z\to Y$ for some $Z$?

Let $X=\operatorname{Spec}(R)$ be an integral scheme with generic point $\eta$ and let $Y$ be a separated scheme. A rational map $X\leadsto Y$ is a certain equivalence class and it is represented by ...
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Problem I.5.4(c) in Hartshorne

The problem asks to show that if $Y$ is a projective curve in $\mathbb{P}^2$ of degree $d$ and $L$ is a line such that $Y \neq L$, then $\sum_{P \in L \cap Y} (L \cdot Y)_P = d$. The solution given ...
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30 views

About the functor between varieties over $k$ and $k$-schemes

Consider an algebraically closed field $k$, Thanks to Hartshorne II(2.6) we know that there exists an equivalence of categories $$F:\textrm{Sch}(k)\longrightarrow\textrm{Var}(k)$$ Where ...
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64 views

blowing-up and tangent cone: essentially identical concepts?

After reading about the concepts of blowing-up and tangent-cone of a curve at a point $P$, i have the following understanding: The blowing-up gives us the slopes of the tangent(s) of the curve at ...
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23 views

Factoring a birational morphism through blowup

Let $X,Y$ be smooth, proper varieties, and $f: X \to Y$ be a proper birational morphism. Suppose $E$ is a smooth, irreducible exceptional divisor, with the image $f(E)$ also smooth. Let $I$ be the ...
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35 views

finite group actions

Let $X$ be a smooth quasi-projective and separated $k$-scheme and $G$ a finite group acting on $X$. Suppose $\mathrm{char}(k)$ does not divide the group order. Then there is the quotient stack ...
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54 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
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Bezout's bound and resultants - reference request

In Terry Tao's blog post about Bezout's inequality, he writes: In our notation*, this theorem states the following: Theorem 1 (Bezout’s theorem) Let $d=m=2$. If $V$ is finite, then it has ...
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1answer
157 views

Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...
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61 views

Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...
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47 views

Generating set of the algebra invariants of finite group.

Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of ...
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1answer
50 views

$\mathbb C$-isomorphism between two $\mathbb C$-schemes.

Consider a field automorphism $\sigma\in\textrm{Aut}(\mathbb C)$, and moreover consider the $\mathbb C$-scheme $p:\mathbb P^1_{\mathbb C}\longrightarrow\textrm{Spec}\,{\mathbb C}$ where $\mathbb ...
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44 views

The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
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19 views

Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
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1answer
38 views

Characterization of the transcendentals over a field

I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page: I couldn't solve any part of the equivalence, I think maybe ...
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26 views

projective change of coordinates and tangent line

Let $C/k$ be a projective algebraic curve given by a polynomial $F \in k[X,Y,Z]$. If $L$ is tangent to $C$ at a point $P$ does $L$ remain tangent to $C$ at a point $P'$ after a projective change of ...
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28 views

Morphisms of quasi-projective varieties

Let $Y\subseteq \mathbb{P}^n(k)$ be a quasi-projective variety. By Görtz, Wedhorn (page 32, Proposition 1.65) in order to show that $$h:Y\to \mathbb{P}^m(k), y\mapsto (f_0(y):\dots :f_m(y))$$ is a ...
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28 views

Intersections on General Nonsingular Projective Varieties

Let $X$ be a nonsingular, integral projective variety of dimension at least 2 over $k$ algebraically closed. Let $Y$ and $Z$ be two codimension 1 subschemes (effective Weil divisors) of $X$. Must they ...
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Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
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Reference request: About Weil book

In "Standard conjectures on algebraic cycles" of Grothendieck and "Algebraic cycles and the Weil conjectures" of Kleiman they say in their references: A. Weil: Variétés Kählériennes, Hermann, ...
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$a,b$ integral $\implies$ $a+b$ integral

I'm sure it's just silly thing. I'm reading Fulton's algebraic curves book and I don't understand this phrase of this proof: I didn't understand why according to the proposition we have $a\pm b,ab$ ...
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50 views

About the isomorphism of two schemes.

Let $C$ and $B$ be two graded $A$-algebras, where $A$ is a commutative ring with unity. Look at the following lemma from Liu's book: Now suppose that $\varphi$ is an isomorphism of graded ...
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103 views

Blow up of base locus of a pencil and line bundles

This is related to a previous question of mine: Is the universal hyperplane section the blowup of the baselocus? Let $X$ be a variety, $L$ a line bundle on it. Take $V < H^0(X,L)$ to be a linear ...
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1answer
36 views

Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?

Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the ...
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147 views

For which $n$ is $\mathbb{A}^n\setminus \{0\}$ affine?

For which $n$ is $\mathbb{A}^n(k)\setminus \{0\}$ an affine variety? I think for $n=0$ and $n=1$ it is. For $n>1$ probably not, but I don't have a proof. $n=0$: Take the ideal $\mathfrak{a}:=(1)$ ...
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What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
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1answer
86 views

How do we know that $f(x)\in Y$?

At page 19 in this book $f:X\to Y$ is defined to be $$f(a):=(\tilde\varphi(T_1')(a),\dots,\tilde\varphi(T_n')(a)).$$ To explain the notation above, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq ...