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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54 views

Relative minimal surface and minimal surface in the algebraic geometry

Liu's book define regular fibered surface X$\to$S is relative minimal surface if it does not contain any exceptional divisor, regular fibered surface X $\to$ S is minimal surface if every birational ...
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33 views

Exercise V.4.2 Shafarevich basic algebraic geometry 2

Find all points of $\operatorname{Spec} \mathbb{C} \otimes_\mathbb{R} \operatorname{Spec} \mathbb{C}$. By definition, $\operatorname{Spec} \mathbb{C} \otimes_\mathbb{R} \operatorname{Spec} ...
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46 views

Frobenius map in scheme theory

Let $\mathbb{K}$ be a field of characteristic $p$ and let $f: \mathbb{A}_{\mathbb{K}}^{1} \mapsto \mathbb{A}_{\mathbb{K}}^{1}$ be the morphism of the form $x \mapsto x^p$. We consider ...
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37 views

Is there a name for $\{(x,y)\mid X\times Y: \phi(x)=\psi(y)\}$, and why is it quasi-projective?

Suppose you have two regular maps $\phi\colon X\to Z$ and $\psi\colon Y\to Z$, which are regular maps of quasi-projective varieties. I have jotted down (from a little informal talk a while ago) that ...
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2answers
35 views

Reform equation for hyperbola in terms of y = f(x)

I am interested in getting the equation of a hyperbola solved for y (for up/down opening hyperbolas). Here is how far I can get for certain General equation: ...
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Blowup of $\mathbb{P}^1\times\mathbb{P}^1$ and a sheaf computation

Let $Y_1=Y_2=\mathbb{P}^1$, $Y=Y_1\times Y_2$, $p_i:Y\rightarrow Y_i$, $i=1,2$ be a canonnical projections and $\pi:X\rightarrow Y$ be a blowup of $Y$ in a finite set of points. How to compute ...
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43 views

Show that the following set is not algebraic.

How can I prove that the set $\{(z, w) \in \mathbb{C}^2; |z|+|w| = 1 \}$ is not algebraic? I just need a hint. Thanks
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1answer
24 views

Tensor product of two f. g. modules vs. algebras

I was wondering if it is different from proving that tensor product of two finitely generated modules is finitely generated versus tensor product of two finitely generated algebras is finitely ...
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1answer
46 views

Is there any algorithm to find the solution of a system of 2 linear and 1 algebraic equation?

I have a system such as: $$\begin{aligned} a_1+b_1t&=u\\ a_2+b_2t&=v\\ a_3+b_3t&=f(u,v)\; \end{aligned}$$ Where a1, b1, a2, b2, a3, b3 are known ...
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79 views

Are birational morphisms stable under base change via a dominant morphism

Let $f: X \to Y$ be a birational morphism of integral schemes and $g: Z \to Y$ a morphism of integral schemes which maps the generic point of $Z$ to the generic point of $Y$, i.e., the morphism $g$ is ...
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1answer
49 views

Resource for coordinate geometry

I am looking for a good resource (preferably in the form of textbooks) for coordinate geometry. Rather than a comprehensive coverage of topics, I am looking more for depth in a particular topic. It is ...
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26 views

Image of polynomial map, what's its coordinate ring?

I'm just trying to get some basic facts straight, Given a polynomial map $$F : \mathbb A^n \to \mathbb A^r,x \mapsto (f_1(x), \ldots, f_r(x))$$ with $f_i \in k[x_1,\ldots,x_n]$, I know that the ...
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1answer
29 views

I have a line F(t) = a + bt and a surface, S(u,v). Is there a formula for the intersection between those?

If I have line, F(t) = a + bt (where $a$, $b$ are known 3D vectors), and a surface, S(u,v), as an arbitrary algebraic formula, ...
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1answer
53 views

Is $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$?

Just curious, is it true that $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$? Here I'm writing $\mathbb{A}^1$ is affine space, and $\mathbb{P}^1$ projective space, both over an ...
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0answers
49 views

Is my intuition on projectivization correct?

Is my intuition on what a projectivization of an affine curve in $C^2$ is and why it is useful correct? From what I understand given an affine curve $C$ we are trying to find a projective curve ...
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29 views

why a projective transormation of a non singular curve gives a non singular cuve?

I was reading a book which has the following exersise.Suppose that V is a nonsingular projective curve and T is a projective transformation. Prove that T(V ) is also a nonsingular projective curve. ...
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27 views

Zariski Topology: Prove that $V(I(X))=\bar{X}$

I am studying Zariski Topology. Here is a problem I am trying to work on: Let $X\subset A^n$ be an arbitrary subset. Prove that $V(I(X))=\bar{X}$. This is my work so far: (1) Since $\bar{X}$ is ...
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1answer
35 views

Morphism of a linear system contracting curves

Let $X$ be a compact complex surface and $L$ a line bundle such that the linear system $|L|$ has no basepoints and $h^0(X,L)>0$. Denote by $\phi:X\rightarrow\Bbb{P}(H^0(X,L)^\vee)$ the morphism ...
3
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1answer
48 views

Borel-Moore Homology and Kunneth Formula

Given two algebraic varieties $X$ and $Y$. It is true that $H^{BM}_n(X\times Y) \cong \bigoplus_{i+j=n} H^{BM}_i (X)\otimes_\mathbb{Q} H^{BM}_i(Y)$. I think that the proof is similar to the one in ...
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1answer
64 views

Sheafs of abelian groups are the same as $\underline{\mathbb{Z}}$-modules

In Vakil's notes page 76, he claims that a sheaf of abelian groups is the same as a $\underline{\mathbb{Z}}$-module, where $\underline{\mathbb{Z}}$ is the constant sheaf associated to $\mathbb{Z}$. ...
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1answer
30 views

Morphisms of the varieties and coordinate rings

I'm thinking something very wrong but I can't find what the flaw in my thinking is. Well, here goes. First, if a contravariant functor $\mathsf{F}: \mathscr{C}\to \mathscr{D}$ is a category ...
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1answer
43 views

Isomorphism of Homs

How can I show the existence of tensor product mappings. Namely, in Liu there is a problems to show that there exists a unique $A$-linear map ...
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1answer
43 views

Comparing vector bundle degrees coming from different embeddings into projective space

This question is a follow-up to this recent question of mine: Comparing notions of degree of vector bundle In that question, the definition of the degree of a vector bundle is discussed — in ...
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84 views

Linear transformation that sends $ax+by+cz=0$ to $x=0$?

I was thinking, can we always find a projective transformation to send any projective line to the line at infinity? After a while I figured that this can certainly be achieved because the projective ...
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1answer
30 views

Determine the radical of an ideal

Determine the radical of the ideal $(x^3-y^6,xy-y^3)$ in $C[x,y]$. I used Nullstellensatz theorem $\sqrt{I}=I(V(I))$. Factorization gives: $$x^3-y^6=(x-y^2)(x+(\frac{1}{2}+\frac{\sqrt{3}}{2}i) ...
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1answer
47 views

Every affine variety in $A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials

I am reading Gathmann's free online notes on Algebraic Geometry. One exercise asks to show that "Every affine variety in $A^n$ consisting of finitely many points can be written as the zero locus of ...
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1answer
39 views

$L$-Zariski closure of subgroup $SL_n(F)$ as subset of $M_n(F)$ also a subgroup of $SL_n(F)$

Let $F$ be a field, and $SL_n(F)$ be the group of $n \times n$ matrices with determinant $1$. Let $\Gamma \subset SL_n(F)$ be a subgroup. We can consider $\Gamma$ to be a subset of $M_n(F) \cong ...
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83 views

What would be the most rigorous book to stydy algebraic geometry and arithmetic curves on my own?

I would like to study algebraic geometry and arithmetic curves on my own but are there suggestions where to start? Namely, I like very rigorous way to do mathematics and I was suggested Liu's book ...
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51 views

Image of a maximal torus via epimorphism

Let $\phi \colon G \rightarrow H$ be a surjective morphism of linear algebraic groups. Let $T \subset G$ be a maximal torus: how can I prove that $\phi(T)$ is also a maximal torus? To show that ...
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15 views

Reduction of general conic

The given equation is - $$3x^2 + 2xy + 3y^2 - 32y +92=0$$ To get rid of xy term i used the substitutions - $$x=p+q , y=q-p$$ Then the equation becomes - $$(p-4)^2 + 2(q-2)^2=1$$ which is an ellipse ...
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1answer
84 views

Proved conjectures (now theorems) in algebra/algebraic number theory/algebraic geometry

I would like to collect some proved conjectures (not so non trivial) in algebra/algebriac number theory/algebraic geometry. For example, I consider Serre's conjecture on projective modules over ...
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1answer
77 views

prerequisites for serre's FAC?

Is the knowledge of undergraduate's basic algebra and general topology enough to reading FAC? Do I need learn some algebraic topology and homological algebra, commutative algebra, or several complex ...
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31 views

Questions about coordinate ring of a closed subvariety.

How to prove the following statement: If $L$ is a closed subvariety of $M$, then there is a surjection from $\mathbb{C}[M]$ to $\mathbb{C}[L]$? I think that maybe this follows from definitions. ...
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2answers
112 views

“Closure” and “neighborhoods” in Spec(A)

While trying to work through the sequence of problems in Atiyah-Macdonald's first chapter regarding the prime spectrum of a ring, I've run across a small point of confusion. Namely: In the point ...
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1answer
74 views

Which polynomial level sets are bounded?

Let $$\begin{align} \mathcal{P}^n&=\left\{p:\mathbb{R}^n\to\mathbb{R}, (x_1,\ldots,x_n)\mapsto\sum_{i_1,\ldots,i_k\in\mathbb{N}\cup\{0\}} a_{i_1\cdots i_k}x_1^{i_1}\cdots x_n^{i_n}\;\bigg|\; ...
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77 views

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
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48 views

Component Group Neron Model Elliptic Curve Cyclic

I'm studying the chapter on Neron Models in Silverman's book "Advanced Topics in the Arithmetic of Elliptic Curves" at the moment, and I do not quite understand why in the split multiplicative case, ...
2
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1answer
49 views

$K_X^*/O_X^*$ is a flasque sheaf for smooth variety over $\mathbb{C}$?

Suppose $X$ is a smooth variety over $\mathbb{C}$, why do we have $K_X^*/O_X^*$ is a flasque sheaf? (Beauville "Complex Algebraic Surface" p.28) (To show the surjection $K_X^*/O_X^*(X)\to ...
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1answer
39 views

Zeros of this multivariate polynomial

I have an equation, wich is somewhat related to the doppler effect : $$ x_1^2x_3^2+x_2^2x_4^2+2x_1x_2x_3x_4-Cx_1^2-Cx_2^2=0 $$ Where C is a known real positive constant. My background in math isn't ...
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35 views

Question regarding the definition of finitely generated graded ring

Let $S = \oplus_{n \geq 0} S_n$ be a graded ring and $S_+ = \oplus_{n \geq 1} S_n$. The notes (Ravi Vakil's online notes on algebraic geometry) I am reading defines the graded ring $S$ is finitely ...
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1answer
187 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
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62 views

What's the connection between exceptional divisor and projectivized tangent space?

This is one homework problem and hence I want some hint but not a whole answer. Let $P$ be a projective space and $X\subset P$ be a non-singular variety. Prove that the collection $L_p$ of lines ...
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1answer
77 views

Comparing notions of degree of vector bundle

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand ...
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2answers
146 views

The Zariski density for two given sets.

Let $A$ and $B$ be two subsets of $\mathbb{C}^n$: $ A = \mathbb{Z}^n$, and $B=\{ (z_1,z_2, \dots , z_n) \in A \text{ such that } z_1>z_2>\cdots> z_n\}$. My questions: Are these two subsets ...
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66 views

Leray's theorem up to some degree

I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology. The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such ...
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2answers
62 views

When a prime ideal is restricted to a basic open subset of projective space, is it still prime?

Suppose $I\subset k[x_0,\ldots,x_n]$ is a prime ideal. Now restricted on the basic open subset $\mathbb{P}^n_{x_i}$ of $\mathbb{P}^n$, is $I$ still prime? Note: 1. Here $\mathbb{P}^n_{x_i}$ is ...
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51 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
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1answer
68 views

Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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1answer
41 views

Equality of two $K$-valued points for reduced $K$

I'm reading "Red Book of varieties and schemes". There is definition 2, page 118. Let $f,g:K \to X $ be to $K-$valued points of scheme $X$, we say that they are equal at $x\in K$ $(f(x) \equiv g(x))$ ...
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1answer
86 views

Is $\mathbb{P}^{1}$ a fine moduli scheme?

I want to show that $\mathbb{P}^{1}_{\mathbb{C}}$ is a fine moduli scheme for the families of lines through the origin of the affine plane. I took a flat family $\mathcal{D}\rightarrow B$ and I tried ...