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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Flat families of semistable sheaves parametrized by $\mathbb{A}^1$.

Suppose we have a non trivial short exact sequence, $$0\longrightarrow F'\longrightarrow F\longrightarrow F''\longrightarrow0,$$ where $F$, $F'$ and $F''$ are semistable sheaves with the same reduced ...
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74 views

Is there an algebraic-geometric solution to the problem of the Leibnizian formalism?

The precise question appears at the end of this entry. With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of $\frac{dy}{dx}$ as ...
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54 views

G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
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63 views

Why is this theorem important in fulton's book?

In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point. afterwards he states this theorem: What'...
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70 views

parametrization of a specific algebraic surface

I consider the surface of degree $5$ of equation: $$ 4 x y - 4 x^3 y - 4 x y^3 + z + 2 x^2 z + x^4 z + 2 y^2 z + 2 x^2 y^2 z + y^4 z - 4 x y z^2 - 6 z^3 + 2 x^2 z^3 + 2 y^2 z^3 + z^5=0 $$ Question: ...
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942 views

What are the prerequisites for Fulton's “Intersection Theory”?

Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?
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86 views

$x^2+y^2=z^2$ in complex numbers

as a prelude to inquiring about solutions of Pythagoras' equation in Gaussian integers, it seemed sensible first to write out this equation for the complex case! i use the notation $z_i=x_i+iy_i$ and ...
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1answer
88 views

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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27 views

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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290 views

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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24 views

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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54 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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1answer
70 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: G\...
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1answer
63 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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1answer
200 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 ...
5
votes
1answer
81 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 ...
3
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0answers
77 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 ...
3
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1answer
55 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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1answer
69 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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93 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 $n^{\times d}$-...
2
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1answer
36 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 ...
4
votes
1answer
101 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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168 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?
3
votes
2answers
140 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, ...
4
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1answer
83 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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1answer
102 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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0answers
40 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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63 views

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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1answer
79 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 ...
4
votes
1answer
134 views

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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1answer
314 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 ...
5
votes
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89 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 ...
2
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51 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 $[X/G]$....
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1answer
61 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 ...
2
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1answer
219 views

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 ...
2
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1answer
202 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 ...
3
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111 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 $\mathbb{...
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1answer
62 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 ...
2
votes
1answer
58 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 P^1_{\...
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1answer
48 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 ...
0
votes
2answers
28 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 ...
3
votes
1answer
49 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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0answers
56 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 ...
3
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1answer
40 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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0answers
50 views

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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1answer
41 views

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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1answer
52 views

$a,b$ 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 $a\pm b,ab$ ...
3
votes
1answer
51 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 ...
7
votes
3answers
339 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 $\mathfrak{a}:=(1)$ ...
5
votes
2answers
213 views

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) {\overset{g}{\...