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
38 views

automorphism of the projective space $\mathbb{P}_A^n$

In exercise 16.4.B of Vakil's notes, he establishes that the group of automorphisms of $\mathbb{P}_k^n$ is $PGL_{n+1}(k)$. This I can manage to show, but in the remarks following the exercise he asks ...
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2answers
57 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
0
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1answer
32 views

Why this application is well-defined?

Let $C$ be a curve. An application $\phi:C\to \mathbb P^n$ is called regular in a point $P\in C$ if there are regular functions $f_0,\ldots,f_n$ defined in a neighborhood $V$ of $P$ in $C$ such ...
0
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1answer
31 views

Compactness of Lie groups

Let $G$ be a Zariski-closed subgroup of $GL(V)$, where $V$ is an $n$-dimensional complex vector space. Question. Does $G$ have the structure of a compact Lie group? Such $G$ certainly is a Lie ...
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0answers
36 views

on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
1
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1answer
13 views

About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
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1answer
18 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
1
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1answer
43 views

Image of an arbitrary map falling on a algebraic set - criterion?

Let $f$ be a "typical" smooth non-polynomial map from $\mathbb{R}^3$ to $\mathbb{R}^7$. Is it reasonable to expect that $f(\mathbb{R}^3)$ is not included within the zero-set of a system of polynomial ...
1
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0answers
43 views

Is restriction of scalars an exact functor?

For the notion of restriction of scalars (aka Weil restriction) I have in mind, see this wiki page. My question then is Is the restriction of scalars an exact functor for ses of smooth linear ...
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0answers
36 views

Picard group schemes of degree d

Let $C$ be a smooth curve. I know that $Pic^0(C)$, i.e. the Picard group of degree 0 line bundles on $C$, is isomorphic to the jacobian $J(C)$, so it is an abelian variety. My question is, what about ...
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40 views

How could we define a sheaf or presheaf of polynomials? [closed]

Good evening everyone , Is there a sheaf or presheaf whose sections are polynomials defined on opens of a topology ? . If yes , what is this topology ?. Is it the Zariski topology , and why? And how ...
1
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1answer
36 views

Dimension and morphism with finite fibers

I'm studying the dimension of projective varieties and in the literature I'm reading I have the following statement: "If $f : X → Y$ is a morphism with finite fibers, i. e. such that $f^{−1}(P)$ ...
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0answers
67 views

Geometric Proof for Fermat's Last Theorem - A Question [closed]

I have been working on a geometric proof for Fermat's last theorem that I just realized has been worked on already in some shape or form (ba-dum-tsh). Before anyone says it, yes, I am aware that this ...
1
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0answers
33 views

algebraic varieties with log terminal singularities

I am looking for some non-trivial examples of algebraic varieties which have log-terminal singularities.
1
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0answers
25 views

Meaning of statement about differential forms

Let $S$ be a complex algebraic surface (smooth and proper over $\mathbf{C}$) and let $D$ be a divisor on $S$. What does it mean for a meromorphic section of the sheaf $(\Omega^1_S)^{\otimes m}$ to be ...
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0answers
27 views

Intersection numbers on product surfaces

Let $C_1$ and $C_2$ be smooth, projective curves over a field $K$. Let $S = C_1 \times C_2$. Let $D$ and $D'$ be (reduced) divisors on $S$ which map dominantly to both $C_1$ and $C_2$. How does one ...
0
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1answer
41 views

$(-1)$-curves and base change along a field automorphism.

Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$. Now consider a field ...
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0answers
19 views

A question in K. Hulek, Elementary algebraic geometry

I'm reading Elementary Algebraic Geometry of Klaus Hulek, and I have a minor question about a proof of Proposition 1.62 in page 48-49. At the end of the proof, $W_i^v = \{w \in W:(v,w)\in Z_i\}$ is ...
2
votes
2answers
74 views

Hartshorne II Prop 6.8

My weaknesses with commutative algebra are really slowing down my progress through Hartshorne. I hope someone can help me understand some statements in the proof of the proposition below. Prop ...
1
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0answers
43 views

A question about an exercise in Klaus Hulek book

"Let $X, Y \subset \mathbb{C}^4$ be varieties defined by $$ X := \{ (t,t^2,t^3,0) \,|\, t \in \mathbb{C} \}, \quad Y := \{ (0, u, 0, 1) \,|\, u \in \mathbb{C} \}. $$ The join variety of $X$ and $Y$ ...
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2answers
22 views

If $\mathcal{I}(-)$ is the ideal map on subsets of affine space, why does $A\subseteq\overline{B}\iff\mathcal{I}(B)\subseteq\mathcal{I}(A)$?

I think this is a basic property of $\mathcal{I}(-)$, but I'm having trouble seeing it. I denote by $\mathbb{A}^n$ the affine $n$-space over an algebraically closed field $k$, where if ...
3
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2answers
26 views

Canonical embedding

I'm reading the second chapter of this paper and I need help to understand what exactly this canonical embedding is: Remark: $C$ is a smooth non-hyperelliptic complete irreducible algebraic curve ...
1
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1answer
22 views

A question about Klaus Hulek algebraic geometry (regarding Noether normalization)

This is the proof of Noether normalization on p.30 of Klaus Hulek's elementary algebraic geometry. And on the next page, the book says that "Analyzing the above proof, we see that y1, .., ym can be ...
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2answers
55 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 - ...
4
votes
1answer
56 views

Hartshorne II prop 6.9

I feel completely in the dark, like I am totally missing what is going on behind the scenes in this section. I apologize in advance. Prop. 6.9: Let $X \to Y$ be a finite morphism of ...
1
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1answer
37 views

Variety of Connected Components

In Milne's text http://www.jmilne.org/math/CourseNotes/iAG.pdf (A71), he introduces the "variety of connected components" of a finite type scheme $X$ over $k$ as the universal example of a zero ...
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0answers
33 views

Smoothness and singularity

How is it possible a algebraic variety be smooth but has a singularity? Smoothness means a variety does not have a singularity?
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2answers
44 views

What is $I(X)$ for $X\subseteq\mathbb{A}^2$ given by $x^2+y^2=x=1$?

Suppose $k$ is algebraically closed, $\mathbb{A}^2$ affine. I'm curious about the subset $X\subseteq\mathbb{A}^2$ given by the unit circle $x^2+y^2=1$ and the line $x=1$. What would the ideal $I(X)$ ...
0
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0answers
24 views

dominant rational map and induced homomorphism injective?

Let V ,W be affine varieties and f: V -> W be a dominant rational map. Then the Klaus hulek book says that it is equivalent to the homomorphism f* : k[W] -> k(V) being injective. I showed if f is ...
0
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0answers
29 views

If $[\omega]\in H^2(M,\mathbb Z)$ then $M$ is projective?

Let $(M,\omega)$ be a Kähler manifold with $[\omega]\in H^2(M,\mathbb Q)$ then why $M$ must be projective variety. As I know if $[\omega]\in H^2(M,\mathbb Z)$ then $M$ is projective by Kodaira theorem ...
0
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2answers
36 views

Definition of homogeneous ideal

I'm a little confused about the definition of a homogeneous ideal. I have the following two definitions: An ideal $I\subset k[X_{0}, \dots, X_{n}]$ is homogeneous if $I$ is generated by (finitely ...
1
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1answer
41 views

Showing that an ideal is not generated by two elements

Let $X=V(x_1, x_2)$ and $Y=V(x_3, x_4)$ be affine varieties on $\Bbb C^4$ where $\Bbb C$ is the complex number. Then, I have to show that the ideal $I(X ∪ Y)$ cannot be generated by two elements. I ...
0
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0answers
30 views

Showing that stalk $O_{Spec \ A, p}$ is $A_p$

Suppose I have $A$, a commutative ring with unity. I would like to show that stalk $O_{Spec \ A, p}$ is $A_p$ for $p \in Spec \ A$. Could someone please explain me how this works? (I am having ...
2
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2answers
34 views

What are some good examples of (non-quasicoherent) sheaves not satisfying the conclusion of Hartshorne Lemma II.5.3?

Hartshorne, Algebraic Geometry, Lemma II.5.3 reads (roughly): Let $X = \operatorname{Spec} A$, let $f \in A$, and let $\mathscr{F}$ be a quasicoherent sheaf on $X$. (a) If $s \in \Gamma(X, ...
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0answers
15 views

Find length of the midpoints of the diagonals in a given trapezoid [closed]

For any given trapezoid, where the bottom base, a, is larger than the top base b -- find the length of MN, the line connecting the midpoints of the diagonals, using only vectors.
1
vote
1answer
26 views

Analytically isomorphic fibers.

Suppose that $S$ is a non-singular complex projective surface with a fibration $f$ over $\mathbb P^1(\mathbb C)$. Suppose also that: There are only finitely many points $y_1,\ldots,y_n\in\mathbb ...
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0answers
28 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
3
votes
1answer
42 views
+50

When will the support of a non-effective Cartier divisor be pure of codimension 1?

Let $X$ be a scheme and $D \in Div(X)$ a non-effective Cartier divisor. I am curious as to when $\text{Supp } D$ is pure of codimension 1, i.e all irreducible components are of codimension 1. So, ...
3
votes
1answer
43 views

Irreducibility of holomorphic functions in a neighborhood of a point

Let $D \subset \mathbb C^n$ be a domain and let $f \in \mathscr O(D)$, $f \not\equiv 0$ be a holomorphic function. Define $$ V_f = \bigl\{ z \in D : f(z) = 0 \bigr\}. $$ Let $p \in V_f$. Suppose ...
0
votes
1answer
62 views

Hartshorne II prop 6.6

I'm having a really hard time understanding the proof of this proposition. $X$ is a noetherian integral separated scheme that is regular in codimension 1. We consider $X\times \mathbb{A}^1$ and the ...
1
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1answer
56 views

Riemann surface from $x^2 + y^2 = 1$ for $x,y \in \mathbb{C}$

I am reading Edward Frenkel's book Love and Math. In Chapter 9, it is talked about the one-to-one correspondence of solution of algebraic function of complex numbers and Riemann surfaces. can anyone ...
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0answers
19 views

A Question from the proof of affine algebraic group is linear

In the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", Prop 1.10): Let $G$ ...
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0answers
27 views

Question regarding a section of an open set of the form $U \cup V$

Suppose I have some scheme $(X, O_X)$. Suppose I have two open subsets of X, $U$ and $V$. I was wondering about the following: 1) Is $\Gamma (U \cup V, O_X) \cong \Gamma (U , O_X) \times_{\Gamma (U ...
0
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1answer
36 views

homeomorphism over zariski topology becomes a homeomorphism over usual topology?

a semicubical parabola $L$ in $\mathbb C^2$ is given by $y^2=x^3$. I showed that a bijective function $f\colon\mathbb C \to L$ defined by $t \mapsto (t^2, t^3)$ becomes a homeomorphism regarding the ...
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0answers
26 views

Projective bundle is projective?

Let $\mathbb{P}(E)$ be a projective bundle over some smooth projective variety $X$, defined over $\mathbb{C}$ for definiteness. Then this bundle is also a smooth projective variety. Smoothness is ...
1
vote
1answer
28 views

What is the example of pseudo-effective divisor which is not an effective divisor

By definition, a pseudo-effective $\mathbb{R}$-divisor is the limit of effective $\mathbb{R}$-divisors in $N^1(X)$, I was wondering what is the example of pseudo-effective divisor which is not an ...
2
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1answer
45 views

A question about Klaus Hulek algebraic geometry

I'm reading Klaus Hulek's algebraic geoemtry and there is something that I can't understand. Here it says that if {p,q} is a counterexample with minimum max{deg p , deg q}, then it can be assumed ...
0
votes
1answer
25 views

Why is a projective subspace itself a projective variety?

Let K^(n+1) be a vector space and W be its subspace. Then projective subspace P(W) is said to be a projective variety of P(K^(n+1)). Could anyone tell me why it is so?
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0answers
28 views

semi-positiveness of canonical line bundle under the condition Kodaira dimension be positive.

Let $M$ be a projective variety with positive Kodaira dimension, then why the canonical line bundle is semi-positive?. Is there any reference?
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0answers
60 views

When is $A = k[x_1,\ldots, x_n]/I$ integrally closed?

Suppose that it is not easy to determine that $A$ is a UFD (or that it is a local, noetherian dimension 1 domain with principal maximal ideal). Can someone suggest strategies for showing that a ...