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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75 views

Show that $V(I \cap J)=V(I) \cup V(J)$.

Let $I$, $J$ ideals of $K[x_1, x_2, \dots , x_n]$. I want to show that $$V(I \cap J)=V(I) \cup V(J)$$ I tried the following: $$\subseteq: $$ Let $x \in V(I \cap J)$. From the definition of $V$: ...
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1answer
60 views

Set of roots of sum is equal to the intersection [closed]

Let $(I_a)_{a \in A}$ be a family of ideals of $K[x_1,x_2, \dots, x_n]$. I want to prove that: $$V \left ( \sum_{a \in A} I_a\right )=\bigcap_{a \in A} V(I_a)$$ Do we have to use the definition: ...
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1answer
44 views

How to decompose into irreducible components

I only know how to find the irreducible components when I know what the image is, but there are lots of equations that are hard to figure out their images, is there any systematic way to find the ...
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1answer
44 views

exercise in Shafarevich

I didn't know what is the meaning of two elements, I just found that xy+xz+yz=0 and xy+xz-yz=0 can satisfy the requirement. Shafarevich--Basic Algebraic Geometry 1 P80
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1answer
37 views

Derivation of the Euler characteristics of the moduli space of rational curves

I am looking for references / hints of proof on the derivation of the Euler characteristics $\chi(\mathcal M_{0,n})=(-1)^{n-1} (n-3)!$ of the moduli space of rational $n$-pointed curves. I have been ...
1
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0answers
29 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
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1answer
45 views

Morphism of varieties $f:X\rightarrow Y$ (Affine)

Im trying to do an exercise from the book Algebraic Curves of Fulton (Exercise $\:6.26^{*}$). It says: Let $f:X\rightarrow Y$ be a morphism of affine varieties. Show that $f(X)$ in dense in $Y$ if ...
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1answer
50 views

How is this method of finding a maximal ideal specific to finite algebras over a field?

Let $A$ be a finitely generated $K$-algebra over a field $K$. A typical problem is to find a maximal ideal $\frak{m}$ such that $f\notin\mathfrak{m}$ and it does not coincide (or contains) another ...
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1answer
74 views

About Zariski topology

Suppose $m,n>0$.Prove that the Zariski topology of $K^n\times K^m$is not equal to the product of the Zariski topologies of $K^n$ and $K^m$. I have thought: Take $ K =\mathbb{C}$ and see the ...
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1answer
34 views

Simple questions about morphisms of finite type and proper morphisms

I'm studying algebraic geometry (it is my first course), by following Mumford's Red Book, and now I'm stacked in some (probably silly) questions about schemes, more precisely in the section 7 of ...
2
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2answers
46 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: ...
3
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0answers
35 views

explicit (holomorphic) map revealing blow-up as a connected sum with $\overline{\mathbb{CP}}^n$

I am trying to understand the statement that a blow-up of a complex manifold $M$ at a point $p$ is equivalent to the connected sum of $M$ with $\overline{\mathbb{CP^n}}$ and, being a physicist, I need ...
5
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0answers
47 views

Real affine variety of $d$ orthonormal vectors in $\mathbb R^n$

I'm interested in the affine variety $$ V = \left\{ \, A\in \mathbb R^{d\,\times\, n} \, \middle| \, A\,A^T = I \, \right\} \subseteq \mathbb R^{d\, \times\, n}, $$ where $n\ge d$ and $I$ is the ...
8
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3answers
86 views

Why do we retain exactness when tensoring by $\mathcal{O}_C$ in Hartshorne, Lemma V.1.3?

Hartshorne, Algebraic Geometry, Chapter V, Lemma 1.3, reads (in part): Throughout this chapter, a surface will mean a nonsingular projective surface over an algebraically closed field $k$. [...] ...
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0answers
39 views

valuation on function fields

Let $p$ be an irreducible polynomial in $k[x]$, for some characteristic 0 field $k$. So we have a valuation $v_p$ corresponding to $p$. Now take $a$ to be a root of $p(x)$. Then $(x-a)$ is irreducible ...
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2answers
54 views

Identities about ideals

If $A,B \subseteq K^n$ show the following: If $A \subseteq B$, then $I(B) \subseteq I(A)$. $I(A \cup B)=I(A) \cap I(B)$ Could you give me a hint, how the above identities could be proven? EDIT: ...
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1answer
27 views

Irreducible components of this variety

Can you give me a clue of how to obtain the irreducible components of $V(y^2 − x^4,x^2 −2x^3 − x^2y + 2xy + y2 − y)$?
1
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1answer
30 views

Is this homomorphism injective?

I want to show that the homomorphism $\mathbb C[x,y]/(x^m-y^n) \rightarrow \mathbb C [t]$, given by $x \mapsto t^n, y \mapsto t^m$ is injective if $n$ and $m$ are coprime. I know that I must show ...
2
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1answer
50 views

Embed $1$-dimensional torus in $SO(2)$

Let $k$ be an algebraically closed field, and let $k^*$ be the one dimesional torus. We want to embed it in $SO(2)$ , the group of matrices $A$ such that $\det A=1$ and $A^tA=Id$. My first attempt ...
5
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1answer
77 views

Differential of a morphism of abelian varieties

I am reading the lecture notes of J.S. Milne on Abelian varieties and I got stuck at some point. Let $\alpha,\beta\colon X\rightarrow Y$ be homomorphisms of abelian varieties $X$ and $Y$. Then for ...
4
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0answers
86 views

Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...
3
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0answers
66 views

Criterion of flatness for projective morphism

Let $f: X \to Y$ be a projective morphism, $\mathcal O(1)$ is a relatively very ample sheaf, then f is flat iff $f_*\mathcal O(m)$ is locally free for big $m$. I can prove flatness implies that ...
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0answers
22 views

$V^T=V(F_1^T,\ldots,F_r^t)$

I'm reading Fulton's Algebraic Curves book on page 19 he defines $V^T$: I want to prove if $V=V(F_1,\ldots, F_r)$, then $V^T=V(F_1^T,\ldots,F_r^T)$, Is this true? I need help Thanks a lot!
2
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0answers
37 views

Problem sets for Griffith and Harris?

Are there any problem sets floating around that correspond to the material in Griffiths and Harris' "Principles of Algebraic Geometry"? I find a good set of problems helps cement the concepts when ...
5
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1answer
74 views

smooth irreducible variety with finite (non-zero) Picard group?

Does there exist a smooth irreducible variety $X/\mathbb{C}$ such that $\mathrm{Pic}(X)$ is finite and non-zero?
0
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0answers
47 views

Coordinate Ring of Disjoint Union of Affine Varieties

Let X and Y be two varieties in affine n-space such that $X\cap Y=\emptyset$. Let $K[X]=K[X_1,...,X_n]/I(X)$ be the coordinate ring of X. I have managed to convince myself that $X\cup Y$ is an affine ...
2
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0answers
49 views

Proof of Riemann-Roch using Mittag-Leffler

In the introduction for Rick Miranda's book "algebraic curves and Riemann surfaces", it says that they will prove Riemann-Roch "in an algebraic manner, via an adaptation of the adelic proof, expressed ...
3
votes
1answer
48 views

Blow-ups followed by contractions

Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly ...
3
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1answer
67 views

Counterexample - modules over non-Noetherian domain

Does anyone know an example of a (necessarily non-Noetherian) domain $A$ and a finitely generated $A$-module $M$ with the property that $M_f$ is not free for any nonzero $f \in A$? This would provide ...
4
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1answer
53 views

Examples of algebro-geometric moduli problems without a “natural” choice of pullback?

When I try to learn about stacks, something that is mentioned all over the place (as a reason not to define moduli problems as functors from schemes to groupoids) is the fact that the pullback of a ...
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0answers
30 views

Composition of morphisms and critical values.

Definition: Let $\varphi:X\longrightarrow Y$ be a morphism between varieties over $k$. We say that $\varphi$ is smooth at $x\in X$ if the following properties holds: $\varphi $ is flat at $x$ . ...
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2answers
46 views

Quick question about a kind of morphism between algebraic varieties

I'm reading a proof where the author needs to use the Stein's factorization theorem. Reading this theorem, i've found the term finite morphism. What does it mean? Can someone tell me a topological ...
2
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1answer
47 views

Vector space and algebraic closure of a field

I hope you can help me with these questions, I can't really come up with a solution! Let $V_k$ be a vector space of dimension $n$ over a field $k$. Let $K=\bar k$ be the algebraic closure of $k$. A ...
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0answers
35 views

Hartshorne Ex. II 1.16 b) Flasque sheaves and exact sequences

the exercise states that when we have an exact sequence $0\to\mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to 0$ of sheaves (say of Abelian groups) over a topological space $X$, and when $\mathcal{F}'$ ...
1
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1answer
73 views

Why is conic isomorphic to $\mathbb{P(C^2)}$?

Given a quadratic form $C(x)=x_1 ^2+x_2 ^2 + x_3^2$ in $\mathbb{C}[x_1,x_2,x_3],$ we have a conic $$C=\{C(x)=0\} = \{[x_1:x_2:x_3]: x_1 ^2+x_2 ^2 + x_3^2 = 0\}$$ in $\mathbb{P(C^3)}$, given in ...
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1answer
24 views

Geometry - Cyclic Quadrilaterals

Three points A,B,C lie on the circumference of the circle, with center as O. If angle(ACB) = 115 deg. Need to find angle (BOC)? Please post your approach?
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0answers
27 views

Is $F(x_1,x_2,\dots,x_n)$ where $(x_1,x_2,\dots,x_n)\in \Delta$,a semi-algebraic function?

Given $$F(t_1,t_2,\dots,t_n)=\int\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}dx_1dx_2\dots dx_n$$ where $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomials whose coefficients ...
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0answers
55 views

question on ideals in rings

Let $S=K[x_1,\dots,x_n]/J$ be a ring where $K$ is a field of characteristic $0$ and $J$ is an ideal with $Z(J)$ being the zero set of the ideal. For every $\tilde{q}\in S$, let $q$ be the lowest ...
3
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0answers
59 views

Describing generators for the fundamental group of an elliptic curve given by an equation

Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a ...
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1answer
100 views

A “trivial” implication I don't understand.

I'm reading the article "Belyi's theorem for complex surfaces - Gabino Gonzalez Diez" and there are few lines of a certain proof that I don't understand (the author claims that all is trivial): ...
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1answer
25 views

Does every quasi-affine variety have an open cover of affine dense subsets?

Suppose you have a nonempty, quasi-affine variety $Y$. Does $Y$ always have an open cover of affine dense subsets? I know that every quasi-affine variety has an open cover by quasi-affine varieties ...
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0answers
63 views

The Theorem on Formal Funtions - Harthshorne Theorem 11.1

Let $f:X \rightarrow Y$ a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$, and let $y\in Y$ be a point. For each $n\geq 1$ we define $X_n=X \times_Y Spec ...
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3answers
64 views

What are the irreducible components of $V(xy-z^3,xz-y^3)$ in $\mathbb{A}^3_K$?

What are the irreducible components of the algebraic set $V(xy-z^3,xz-y^3)$ in $\mathbb{A}^3_K$? Here I"m just letting $K$ be an algebraically closed set. Normally, what I do is take the equations ...
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1answer
36 views

Orbits of a Torus action

I'm sorry if my questions are trivial, but i can't solve them 1) Let $X$ be an irreducible affine variety with the action of a torus $T$. Is it always true that there exists a dense orbit for the ...
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0answers
32 views

Number of base points of a linear system on a surface

I use Hartshorne's notations: Let $S$ be a non-singular complex surface and moreover let $D$ be a (Weil) divisor of $S$. Now consider a linear system $\delta\subset |D|$; we say that a point $p\in S$ ...
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1answer
54 views

Proof of the existence of Lefschetz Pencils.

Let $S$ be a smooth complex projective surface. A Lefschetz pencil over $S$ is a rational map (which is not a morphism) $f:S--\rightarrow\mathbb P^1_{\mathbb C}$ with the following property: All but ...
3
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2answers
75 views

Does existence of non-trivial solution of $S(x,y,z) = 0, \; S(y,z,x) = 0, \; S(z,x,y) = 0$ implies existence of trivial solution at $x=y=z$ axis?

My question is following. Suppose that you have an implicit surface given by equation $S(x,y,z) = 0$ (if it matters, now $S(x,y,z)$ is a polynomial function). I'm interested only in $\mathbb{R}^3_{+}$ ...
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1answer
34 views

Group law, cubics and Lie group

Let $C$ be a smooth complex cubic in $CP^{2}$. We know that there is a group structure by using the intersection of projective lines (cf. Ried, Undergraduate AG, Section 2), which is really different ...
14
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2answers
279 views

What is the most influential work of Grothendieck in mathematics?

Recently Alexander Grothendieck has passed away but his mathematical wave is still alive and passes its growth ages. It is hard to describe the influence of such a great man in mathematics just in few ...
2
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1answer
50 views

Non-injective affine quasi-coherent module induced by injective module over the global sections??

Recently, I came across this answer on MO, which (allegedly - I have trouble understanding EGA/SGA, so I have not checked the reference) provides a reference for an injective $A$-module, $A$ a ...