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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21 views

Definition of $\sum_{a \in A} I_a$

If $(I_a)_{a \in A}$ a family of ideal of $K[x_1,x_2, \dots, x_n]$, I have the following definition in my notes: $$\sum_{a \in A} I_a=\{ a_{i1}+a_{i2}+ \dots+ a_{ij} | a_{ij} \in I_{a_j} \}$$ Is ...
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1answer
36 views

How to prove that $CP^4$ cannot be immersed in $R^{11}$

Please let me know how to prove that $CP^4$ cannot be immersed in $R^{11}$. I know a proof using an integrality theorem for differentiable manifolds but I want to know if a more direct and simple ...
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1answer
21 views

Global sections of canonical sheaf and genus

Let $k$ be algebraically closed and C a curve. Then $g(C) = h^0(C, \omega_C)$. I want to show that if all the degree $0$ line bundles are trivial, then $g = 0$. I can sort of see that if I think of ...
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1answer
32 views

Definition of a Elliptic curve

I've seen two different definitions of an elliptic curve. 1) The first one being that it is a nonsingular projective curve of genus 1. 2) The other definition nonsingular projective curve of ...
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22 views

Blowing-up of a linear sub-space is Fano?

Consider $P=\mathbb{P}^3\times \mathbb{P}^1$ with coordinates $([x_0:x_1:x_2:x_3],[u:v])$ and let $\varepsilon:X\to P$ be the blow-up of $\mathbb{P}^2\cong A=(x_3=v=0)\subseteq P$, of pure codimension ...
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1answer
26 views

Why a semi-stable non stable bundle $E$ is S-equivalent to $L_1\oplus L_2$

Let $M(2,d)$ be the set of all vector bundles of rank $2$ and degree $d$ over a smooth projecitve curve of genus $g\geq 3$. Let $M(2,0)^s$ and $M(2,0)^{ss}$ be the stable and semistable vector ...
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1answer
38 views

Is a flat coherent sheaf over a connected noetherian scheme already a vector bundle?

Let $A$ be a connected noetherian ring (not necessarily irreducible), $M$ be a finitely presented flat $A$-module. Then $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for each $\mathfrak{p} ...
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1answer
22 views

Why does the Hecke correspondence preserve principal divisors?

Let $p$ be prime not dividing $N$. Consider the Hecke correspondence $T_p$ inducing a set valued function on $X_0(N)$. I'd like to understand why it acts on $\text{Pic}^0$, and so I'd like to know why ...
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1answer
33 views

Exercise of commutative algebra, rational functions.

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints. ...
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1answer
42 views

Does it hold $\mathcal{O}_X (U) =\bigcap_{x \in U} \mathcal{O}_{X, x} \in K (X)$?

maybe this is a stupid question, but I'm not seeing if this is true for some spaces (affines at least). Let $K (X) = \varinjlim\limits_{\emptyset \neq U \in \text{Open}(X)} \mathcal{O}_X (U)$ be the ...
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33 views

Domain of definition of $(1-y)/x$ on $x^2+y^2=1$?

I'm self-teaching myself some basic algebraic geometry, and I wanted to double check something that seems too easy. An exercise sheet I found asks to compute the domain of definition of the rational ...
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24 views

Question about dimension in Notherian spaces

Let $X$ be a Notherian topological space of finte dimension which is Kolmogorov (meaning that for two points $X$ there exists an open subset of $X$ containing one of them but not the other). This ...
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45 views

Sheaf cohomology of certain line bundle

First of all I am sorry if this question is duplicate. I tried to search it but didn't succeed. Let $X$ be a blowing up of projective plane $\mathbb{P}_k^2$ at one point and $L$ be an exceptional ...
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50 views

Hartshorne 3.5.3, how to use surjectivity on global sections and Nakayama to conclude globally generated

There is a step in 3.5.3 of Hartshorne that I am stuck at. The setup is this: Let $\mathcal{F}$ be a coherent sheaf on a scheme $X$ which is proper over $\text{Spec}(A)$ for a noetherian ring $A$. ...
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20 views

Homogeneous polynomials and line bundles

In Huybrechts' Complex Geometry text, he makes the following claim: (Claim) "Any homogeneous polynomial $s \in \mathbb{C}[z_o,\dots, z_n]_k$ of degree $k$ defines a linear map ...
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1answer
35 views

What is The Output of $(a^x*b^{-x/2})(a^{x-y}*b^y)$?

I was solving some problems when I came across a "slick" way to solve these types of problems, and I substituted different numbers for the variables and the answer has something to do with ...
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1answer
27 views

A condition that an algebraic set is irreducible.

From the book by Kenji Ueno, Algebraic Geometry 1. From Algebraic Varieties to Schemes: "If an algebraic set $V(J)$ is reducible, it can be expressed as: $$(1.8)\quad V(J)= V(J_1)\cup V(J_2), \ ...
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22 views

For a ring homomorphism, why does $f$ induces a homeomorphism from $SpecB$ onto the closed subset $V(\ker f)$ of $SpecA$.

Let $\varphi : A \rightarrow B$ be a ring homomorphism. Then we have a map of sets $Spec(\varphi):Spec(B) \rightarrow Spec(A)$ defined by $p \mapsto \varphi^{-1}(p)$ for every $p \in SpecB$. ...
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22 views

Is this equality of varieties valid?

In my book, Iam reading about the Zariski topology on varieties, however I read that the union of the varieties is again a variety and $$V(I) \cup V(J) = V(I \cap J)=V(IJ)$$ this is from Miles Reid, ...
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23 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant over a curve of genus $g$ [duplicate]

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
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1answer
28 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
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1answer
48 views

Transcendental solution to system of equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
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20 views

A question on vectors represented by multilinear polynomials

Let the set of multilinear polynomials of degree atmost $t$ in $\Bbb Z[x_1,\dots,x_n]$ be $\Bbb Z^{t}[x_1,\dots,x_n]$. Let $S=\{0,1\}^n$. Fix an ordering of $S$. For every $f\in\Bbb ...
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1answer
20 views

A sufficient condition for irreducibility of a $G$-variety

Let $G$ be an algebraic group over a field $k$ and let $V$ be a variety on which $G$ acts. Suppose $U\subset V$ is a closed, irreducible, $G$-stable subset which intersects every $G$-orbit ...
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1answer
28 views

Linearly equivalent divisors and linear transformations

Let $C$ be a projective nonsingular irreducible curve. Let $D$ be a divisor on $C$. Suppose $l(D) = n > 0$ and $L(D) = \langle f_1, \ldots, f_n \rangle$. Consider the map $\varphi_D : C \to ...
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1answer
51 views

Cl(Spec A) = 0 implies A is a UFD

This is bothering me and I'd appreciate any clarification. If I know that $ \text{Cl(Spec $A$)}= 0$, why doesn't that imply that $A$ is a UFD? Why do I need that $A$ is integrally closed? This comes ...
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1answer
29 views

Showing that the image of a morphism is algebraic curve

I'm stuck at the first part. I think the image of this morphism should be described by three polynomial equations. I think I found one, which is xw-yz=0 where (x, y, z, w) is the coordinate of K^4. ...
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33 views

Computing the dualizing sheaf of an exceptional divisor

Let $D_1=Z(f_1)$ and $D_2=Z(f_2)$ be two plane cubics meeting in nine points $p_1,\dots,p_9\in\mathbb P^2$. The $f_i$'s induce a rational map $$(f_1,f_2):\mathbb P^2\dashrightarrow\mathbb P^1$$ ...
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1answer
21 views

$J \subset I(V(J))$ where $J$ is an ideal.

The textbook says it's by definition, but as I see it the inclusion should be reversed should it not? I mean $I(V(J))= \{ f\in k[x_1,\dots, x_n]: f(a_1,\dots ,a_n)=0 \text{ for an arbitrary element } ...
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32 views

Alternative proof of Noether Normalization Lemma

On Mumford's Red Book (end of section 7 of chapter 2, pg 126, 127), there is an alternative proof of Noether's Normalization Lemma that goes like this: For an affine variety $X$ over an algebraically ...
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18 views

What does it mean to restrict a function germ to a set germ?

Two sets $S$ and $T$ define the same germ at a point $\xi$ in a topological space $M$ if there is a neighbourhood $U$ of $\xi$ such that $S \cap U = T \cap U$. Two functions $f,g : M \rightarrow \Bbb ...
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1answer
24 views

Identifying lines in $\mathbb P^2$

Let $L$ and $M$ be two lines in $\mathbb P^2$. Does there exist a map $f : \mathbb P^2 \to X$ that "identifies" $L$ and $M$, in the sense that $f\vert \mathbb P^2 \setminus (L \cup M)$ is an ...
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1answer
84 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
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21 views

Tensor product of local Artinian rings

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that ...
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2answers
238 views

Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ ...
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32 views

27 lines on Fermat cubic

Fermat cubic is $S=\{(x:y:z:w)|x^3+y^3+z^3+w^3 =0\} \in \mathbb{P}^3$. It is obvious that 27 lines on Fermat cubic are represented by $(x,ax,z,bz)$ for cube root $a,b$ of $-1$ and their conjugates. ...
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1answer
39 views

Very ample divisors and the Riemann-Roch theorem

What is the easiest way to prove that a divisor $D$ is very ample if and only if $l(D - P - Q) = l(D) - 2$ for all points $P, Q \in C$. It seems like it might be a consequence of the Riemann-Roch ...
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1answer
30 views

An algebraic curve $C$ as a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$

Could someone explain what it means for an algebraic curve $C$ to be a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$, ramified over $n$ points?
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1answer
26 views

singular point of a complete intersection surface

Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$. ...
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1answer
35 views

Subjects or recent progress in Tropical geometry or similar suitable for undergraduate investigation

I'm worried this might not be fitting for this forum, but it's basically a literature and reference request. I'm looking to do a project in algebra where we are supposed to research some topic ...
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27 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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31 views

non-abelian Galois cohomology

Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G ...
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1answer
67 views

About Plücker embedding

I'm doing a work about Plücker embedding and I need some help about a few topics. I'm going to list them: $1-$ I know that Plücker embedding is well-defined and is injective. However, Plücker ...
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1answer
50 views

What is the canonical morphism of $\mathbb{P}^n_A$ to $\text{Spec }A$?

On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$. Later, he ...
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1answer
47 views

Proof of Nike's trick: Two affine open subsets contain a simultaneously distinguished open subset

I'm trying to work through this proof of Nike's tick. Statement of the lemma: Let $ U_{i} = Spec\ A_{i} $ for $ i\in\{1,2\} $ be two open affine subschemes of a scheme $ X $. For $ x\in U_{1}\cap ...
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42 views

Is the image of a morphism between affine schemes always constructible?

Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed ...
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1answer
48 views

$I$-smoothness in Algebraic Geometry

I was reading in Chapter 10 of Matsumura's book about $I$-smoothness. In the book, the autor defines this concept by the following universal property: Let $A$ be a ring, $B$ an $A$-algebra and $I ...
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2answers
17 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...
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42 views

Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
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1answer
33 views

Global sections of the projective space

Let $k$ be an algebraically closed field, and let $\mathbb{P}^n_k=\operatorname{Proj}(k[x_0,x_1,\dots,x_n])$, with structure sheaf $\mathcal{O}$. I would like to know how to prove that ...