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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1answer
179 views

Divisor associated to a rational map

Let $C$ be a smooth projective curve, and let $\phi:C\rightarrow\mathbb{P}^1$ be a non-constant rational map. How to recover a divisor $D$ of the curve associated to this rational map? (I understand ...
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1answer
49 views

Odd-degree hyperellptic curve ramified at infinity

Let $C$ be the projective curve with the affine model given by the equation $Y^2=F(X)$, where $F$ is a polynomial in $x$ with degree $d$ over a field $k$ with characteristic not equal to 2. When $d$ ...
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1answer
72 views

Equation of a quotient variety

We define the variety $V$ as $\{ (x,y,z) \in \mathbb{A}^3\ |\ x = yz \}$. On this variety, I can make $(\mathbb{C}^*)^2$ act by $(\lambda, \mu) \star (x,y,z) = (\lambda \mu\ x , \lambda\ y,\mu\ z)$ on ...
2
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1answer
219 views

homogeneous polynomials as sections of line bundle, surjectivity of multiplication

The homogeneous polynomials in $\mathbb{C}[x_0,\cdots,x_n]$ can be considered as the global sections of a line bundle over $\mathbb{P}^n$ (the line bundle corresponding to Serre's twisting sheaf). In ...
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0answers
182 views

Equivalent conditions to a rational map $f$ being birational

I'm going to copy out a chunk of Hulek's 'Elementary Algebraic Geometry' (page 83). Let $V$ and $W$ be irreducible quasi-projective varieties. Theorem 2.49 For a rational map $f : V \to W$ ...
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402 views

Computing the “lying over”, “going up”, “going down” ideals.

For any commutative unital ring $R$ and an ideal $\mathfrak{a}$ of $R$, we shall denote $$\begin{align*} \mathrm{Spec}(R)&:=\{\text{prime ideals of }R\},\\ ...
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93 views

Query about definition of birational map

Let $V,W%$ be irreducible quasi-projective varieties. A rational map $f : V \to W$ is called birational if there is a rational map $g : W \to V$ with $f \circ g = \mathrm{id}_W$ and $g \circ f = ...
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41 views

a technique of parameterization fixing a point P a taking lines through P

Maybe someone knows this technique. Given $ F\left( {x,y} \right) \in {\Bbb C}\left[ {x,y} \right] $, let $ C = \left\{ {\left( {x,y} \right) \in {\Bbb C};F\left( {x,y} \right) = 0} \right\} $. ...
2
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1answer
191 views

Pullback map on differential forms

Let $X$ and $Y$ be varieties, and let $\phi:X\rightarrow Y$ be a regular map. Let $x\in X$, and $y=\phi(x)$. Write $\Theta_{X,x}$ and $\Theta_{Y,y}$ for the respective tangent spaces of $X$ and $Y$ at ...
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286 views

Localizations of quotients of polynomial rings (2) and Zariski tangent space

I am sorry, in the whole text below $k$ is just meant to be $\mathbb{C}$. This question is closely related to my previous one here. I am considering the two rings $k[X]=k[x,y,z]/\langle ...
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55 views

What are the branch points of $X(n)\to X(1)$

Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps). ...
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2answers
182 views

Projective morphism, generated by global sections basic question

I have a very dumb question. Let $X = \mathbb{P}^2_k = Proj(k[x,y,z])$ where $k$ is algebraically closed. We have an invertible sheaf $\mathcal{O}(2)$ on $X$. Its space of global sections contains ...
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1answer
69 views

A question about Denseness in $\mathbb{P}^n$

$$ \mathbb{P}^n=\bigcup_{j=1}^{n} U_j,$$ where $$U_j=\{[x_0:\dots,x_n]\in \mathbb{P}^n : x_j\neq 0\}=\{[x_0:\dots,x_n]\in \mathbb{P}^n : x_j=1\}$$ can be identified with $\mathbb{C}^n.$ In fact we ...
3
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1answer
363 views

Projective Nullstellensatz

I'm confused about the proof of the Nullstellensatz for projective varieties. If $J \subset k[x_0, \ldots , x_n]$ is a homogeneous ideal, we may regard $V(J)$ as a closed subset $ V(J) = V \subset ...
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2answers
137 views

Projective varieties are the zeroes of finitely many homogenous polynomials

Define a projective variety to be a subspace $V \subset \mathbb P^n$ such that $V$ is the zero set of some set $T$ of homogenous polynomials in $k[x_0, \ldots , x_n]$. My book claims that "as with ...
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122 views

For $X$ an affine variety, $X \setminus Z(f)$ is isomorphic to an affine variety

Let $X \subset \mathbb A_k^n$ be an affine variety, and let $Z_f = X \setminus Z(f)$. I'm looking at a proof of the fact that $\mathbb Z_f$ is isomorphic to an affine variety. The proof proceeds as ...
5
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1answer
293 views

What is the importance of the Krull's principal ideal theorem

What is the importance of the Krull's principal ideal theorem in later study of commutative algebra and algebraic geometry? Can any one tell me the geometric picture of this theorem? Thank you very ...
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418 views

Why are affine varieties except points not compact in the standard topology on $C^n$ ?

I am starting to learn algebraic geometry and in the notes I am reading there is the following remark: " Over the complex numbers and with the strong topology we see that $A^n$ and affine varieties ...
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62 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
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78 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
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94 views

Moving divisors on a surface

I'm trying to understand the proof at the top of page 3 of http://math.stanford.edu/~vakil/02-245/sclass6A.pdf http://math.stanford.edu/~vakil/02-245/sclass6B.pdf Why can $D$ and $D'$ be moved ...
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203 views

Divisor on surface

I'm trying to understand the following result: Let $S$ be a smooth, projective surface over $\mathbb{C}$ and let $D$ be a divisor on $S$. Let $H$ be a hyperplane section of $S$ for a given embedding. ...
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347 views

Intersection number & non-singular curves

I was looking at a proof of the following theorem... Let $S/\mathbb{C}$ be a smooth projective surface, let $C$ be a non-singular irreducible curve on $S$. Then for all $L \in \text{Pic}\,S$, the ...
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1answer
110 views

Generically finite morphism of surfaces

Let $S_1$ and $S_2$ be smooth projective surfaces over $\mathbb{C}$ and let $f: S_1 \to S_2$ be a morphism which is generically finite of degree $d$. How does one prove that $f_* f^* D = dD$ for all ...
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104 views

Projective varieties: Reduction to the affine case

I ran across a certain type of argument for the second time now. Assume that $f$ is some rational map between projective varieties $X$ and $Y$, then supposedly I may replace $X$ with an open affine ...
8
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1answer
593 views

Affine scheme $X$ with $\dim(X)=0$ but infinitely many points

As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ...
2
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1answer
185 views

Conjugacy classes in GL(n)

Given an element $\gamma$ in $GL(n,F)$, where $F$ is either a global field or a non archimedean local field. Assume $\gamma$ is elliptic, i.e. its characteristic polynomial irreducible. Let $Z(F)$ be ...
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194 views

Pullback and pushforward

Say I have a scheme $X$, irreducible and of finite type over a field $k$, and a closed subscheme $Y$ of $X$ with associated closed immersion $i: Y \to X$. Consider a sheaf $F$ on $X$ (for the étale ...
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1answer
257 views

Canonical divisor of an abelian variety

Let $A$ be an abelian variety over a field $k$ and let $K_A$ be its canonical divisor. Then I'm almost certain that $K_A$ is trivial, but I can't seem to prove it, nor find a counter example, nor ...
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1answer
208 views

How to prove that a linear system is *not* base point free?

Assume I have a line bundle $\mathcal{L}$ on a projective, nonsingular variety $X$ over, say, an algebraically closed field (in fact, you may assume $\mathbb{C}$). Given global sections ...
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77 views

Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I ...
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1answer
126 views

Show that (vector) subspaces of $\mathbb{A}^n$ are algebraic sets

i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of ...
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1answer
702 views

Connected components of a scheme are irreducible

Update 2: I posted an answer to this question. Update 1: Problem is now solved because of the excellent hint by Qil. So, if someone wants to post an answer just for the sake of closing this question ...
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1answer
140 views

Does the absolute Galois group act on the moduli space of curves

Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of ...
3
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1answer
102 views

For curves, is being defined over a number field invariant under birational equivalence

Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers. Does this imply that $X$ itself can be ...
2
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1answer
132 views

Another question on scheme morphisms

I have two questions on scheme morphisms. Is the property of a scheme morphism to be a closed immersion a local property (as it is for open immersions)? Let $X=Spec (R)$ be a noetherian scheme and ...
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1answer
171 views

extending Zariski closed sets

Let $U \hookrightarrow X$ be an embedding of algebraic varieties such that $U$ is dense in $X$. Then any Zariski closed subset of $U$ is a trace of a Zariski closed subset of $X$. It escapes me why ...
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1answer
202 views

Irreducible algebraic sets

Let $K := (uw - v^2, w^3 - u^5)$. Show that $V(K)$ consists of two irreducible components, one of which is $V(uw - v^2, w^3 - u^5, u^3-vw) = V(J)$. I don't know how to start this. I see that $V(K)$ ...
6
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1answer
90 views

Example of different schemes with the same space and stalks

What is an example of two non-isomorphic schemes $(X,\mathcal{O}_X)$ and $(X,\mathcal{O}_X')$ with the same topological space such that there are isomorphisms $\mathcal{O}_{X,p}\cong ...
3
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0answers
43 views

Expressing projective varieties in terms of matrix rank

Consider the map $\phi : \mathbb P_k^1 \to \mathbb P_k^3$, where $ \phi(t_0:t_1) = (t_0^3 : t_0^2 t_1:t_0t_1^2:t_1^3)$. Apparently, the image $C := \phi(\mathbb P_k^1)$ is a projective variety given ...
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264 views

Explaining the motivation behind two different definitions of a generic point

This question is primarily regarding the definition of a generic point of a topological space that I came across in Qing Liu's Algebraic Geometry and Arithmetic Curves. First I will give the ...
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2answers
165 views

Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that ...
3
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1answer
58 views

Morphism $f : \mathbb C^n \to \mathbb C^m$ whose image is not algebraic?

How can I construct a polynomial function $f : \mathbb C^n \to \mathbb C^m$ whose image is not algebraic? I can't really get anywhere here. Any hints would be greatly appreciated. Thanks
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1answer
179 views

Intersection of the Irreducible Components of Intersections of Schubert Varieties

Let $K$ be an algebraically closed field and $G$ be the Grassmannian of $k$ planes in some $l$ dimensional vector space $V$ over $K$. Let $V_1\subsetneq ... \subsetneq V_l$ be a flag for $V$. A ...
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2answers
52 views

Ideals and subideals

Let $J = (uw -v^2, u^3 - vw, w^3 -u^5) \subseteq \mathbb{C}[u,v,w]$ and let $I = (uw -v^2, u^3 - vw,w^2 - u^2v) \subseteq \mathbb{C}[u,v,w]$ Show $J \subset I$. To me, it doesn't seem like this ...
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1answer
114 views

Why is the domain of a rational function necessarily nonempty

Let $V$ be an irreducible affine variety. A rational map $f : V \to \mathbb A^n$ is an $n$-tuple of maps $(f_1, \ldots , f_n)$ where there $f_i$ are rational functions i.e. are in $k(V)$. Th map is ...
3
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2answers
500 views

Unions and intersections of algebraic varieties

Let A = $k[x_1, x_2, \ldots, x_n]$ and let $I_{\lambda}$ be an ideal of A. Let J = $\sum_{\lambda \in \Lambda} I_{\lambda}$ be a finite sum. Show that $V(J) = \cap_{\lambda \in \Lambda} ...
6
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1answer
392 views

Prop. 2.3 Hartshorne: $\varphi:A\to B$ induces a morphism $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$

I don't fully understand a step in the proof of the above-mentioned Proposition; more precisely, in part (b): If $\varphi:A\to B$ is a homomorphism of rings, $X=\operatorname{Spec}(A)$, ...
7
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1answer
336 views

Does a regular function on an affine variety lie in the coordinate ring?(Lemma 2.1, Joe Harris)

I think the proof in for Lemma 2.1 in Joe Harris's book Algebraic Geometry, A First Course, does not work. (The statement is on Page 19, and the proof on Page 61.) The proof fails because that ...
3
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1answer
149 views

Hilbert function on ideal generated by linear forms.

This is a slight extension of a remark a read a few days ago. Let $K$ be a field, and let $A=K[X_0,\dots,X_N]$ be a polynomial ring, which is graded in the standard way (the elements of degree $n$ ...