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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26 views

What are the linear equivalence relations being talked about in the definition of a Picard group?

These set of notes say the following: The set of divisors Div$(X)$ on a compact Reimann surface $X$, modulo linear equivalence relations, forms a group Pic$(X)$. What are these "linear ...
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39 views

Example of two affine varieties $X,Y$ such that the image of $\phi:X \rightarrow Y$ is not locally closed

In my course Algebraic Geometry I always find it hard to come up with examples or counterexamples. For instance in the following question: Give an example of two affine varieties $X,Y$ and a morphism ...
3
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0answers
36 views

Arithmetic genus of divisors on cubic surface

This is a question from Hartshorne's Algebraic Geometry (Chapter V.4). It asks to show that for any divisor $D$ of degree $d$ on a smooth cubic surface $X$ in $\mathbb{P}^3$ the following inequality ...
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2answers
136 views

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
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1answer
40 views

Why does a linear equation define a point-set of dimension one less than the space?

For example: If we are in 2-space (2 unknowns), a linear equation defines a line. If we are in 3-space (3 unknowns), a linear equation defines a plane. I mean, it seems obvious, but an explanation ...
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44 views

Associating to every line a vector in that line in an algebraic way

Let $X$ be a complex variety and $\mathscr E$ a locally free sheaf on $X$. Consider the fiber bundles $$ \mathbb P(\mathscr E) \overset{def}= \mathrm{Proj}(\mathrm{Sym}(\mathscr E)), \quad \mathbb A(\...
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3answers
1k views

Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived ...
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2answers
39 views

is an open subset of a noetherian connected topological space connected?

Let $X$ be a noetherian connected topological space $X$. If $U$ is an open subset of $X$, must $U$ be connected? This is not true if $X$ is not noetherian e.g take $X=\mathbb{R}$ in the usual topology....
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2answers
56 views

Homotopy group of the conformal group

I would like to know which are the first three homotopy groups of the conformal group SO(4,2): $$ \pi_n(SO(4,2))=? \quad n=1,2,3 $$
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32 views

component of a divisor is mobile

If D is a mobile divisor, why there exists a divisor $D'\in |D|$, such that any component of $D'$ is mobile? Somebody told me that it follows from Bertini Theorem. Can anybody give me some details?
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39 views

general member of a linear system

Let $|L|$ be a linear system of a quasi protective variety X, and Z a subvariety of X. If Z is not contained in BS($|L|$), is it true that Z is not contained in a general member of $|L|$ ?
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16 views

Two collineations

Give collineations to prove the following (in the extended projective plane): a, One cannot contruct the midpoint of two points using a straightedge only. b, One cannot construct the reflection of a ...
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0answers
27 views

Is $A^2 - (0,0)$ quasicompact?

Is $\mathbb{A}^2 - (0,0)$ quasi-compact (in the Zariski topology)? Surely this is well known. I ask because then it would give an example of a quasi-affine scheme that is not affine.
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0answers
38 views

What *is* the coordinate ring K[V] of an algebraic variety V?

I've been trying to understand this for a while. If I understand it, we let V be an algebraic variety (set?) then define I(V) to be the ideal generated by V. The coordinate ring is K[V] = K[X]/I(V), ...
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0answers
58 views

Let X and Y be irreducible varieties. Show that if X is projective,the projection X × Y →Y induces an isomorphism on the rings of regular functions.

How to prove the exercises above? And I don't know what is the isomorphic actually means between $O(Y)$ and $O(X×Y)$
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0answers
31 views

How to fix this proof that isomorphic varieties have the same dimension? Is it possible?

I am trying to prove the following: Show that affine algebraic varieties that are isomorphic have the same dimension. For completeness let's state the definitions: Let $V,W$ be varieties. ...
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0answers
38 views

Solution verification: Curve is given by points $(t,t^2, t^3)$

I tried to solve the following exerice: Show that the twisted cubic curve corresponding to the affine variety $V(x^2 - y)\cap V(x^3 - z)$ consists of all points in $\mathbb A^3$ of the form $(...
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4answers
170 views

Where can I learn about differential graded algebras?

I want to learn more about differential graded algebras so that I can construct explicit examples of derived schemes over characteristic 0, compute smooth resolutions of morphisms of schemes, and ...
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0answers
30 views

How to find the Coordinate equation of a curve which bends all the parallel rays from infinity towards a single point

How should I proceed on to find the coordinate equation of a curve such that it bends all the parallel rays coming from infinity towards a single point. Yes I know that it would be a 2nd degree ...
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0answers
39 views

Picard number of Kahler manifold

Let $(M,\omega)$ be a Kahler manifold. How can we define simply the Picard number for the special case where $M$ is also projective? Wikipedia defines it as the rank of the Neron-Severi group. In ...
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0answers
55 views

adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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1answer
61 views

How to resolve the singularity of $xy+z^4=0$?

This singularity can not be resolved by one time blow-up. I don't know how to blow up the singularity of the "variety" obtained by the first blow-up, in other words, I am confused with how to do the ...
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0answers
42 views

sequence of cohomology groups associated to Koszul complex

In the paper "On branched coverings of some homogeneous spaces" of Kim and Manivel one reads, that there is a Koszul complex associated with a section $s$ of a locally free sheaf $S$ of rank $p$ (in ...
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0answers
35 views

Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
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1answer
35 views

$k[t]$ is finitely generated $k[x,y]/(y^2-x^2-x^3)$ -module

I am reading example 3, section 7.3 of vakil's notes. It says that $k[t]$ is a finitely generated $k[x,y]/(y^2-x^2-x^3)$ -module by 1,t. This really confuses me. And he also claim $D(t^2-1)$ is ...
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0answers
25 views

"Correct'' morphism extension to Nagata compactifications

Can a morphism of separated schemes of finite type over a field be extended to Nagata compactifications of the schemes preserving the closed complements? Let $\mathbf{Sch}/k$ be the category of ...
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2answers
55 views

Exercise 1.11 Harris Algebraic Geometry: A First Course

I am trying to do part (b) of Exercise 1.11 in Harris' book Algebraic Geometry: A First Course. Let $F_0=Z_0Z_2−Z_1^2$, $F_1=Z_0Z_3−Z_1Z_2$, $F_2=Z_1Z_3−Z_2^2$ (s.t. $V(F_0,F_1,F_2)$ is the twisted ...
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0answers
25 views

ADE classification of singular surfaces (catastrophe theory)

I have seen a lot the Arnold's classification of singular surfaces by the simple Lie groups. I have even asked the author of a book that used this classification about its origin and his answer was ...
2
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1answer
42 views

Is there a projective morphism from the quadric surface to the projective plane with degree 1?

Is there a projective morphism from the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ to the projective plane $\mathbb{P}^2$, with degree $1$?
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1answer
52 views

Is a hypersurface really defined by an arbitrary polynomial?

In An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...
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1answer
83 views

Cohomology Class of a Subvariety

I'm working on question 7.4 of Chapter III.7 in Hartshorne's Algebraic Geometry. The question is about the cohomology class of a subvariety. The setup is as follows: $X$ is an $n$-dimensional non-...
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1answer
20 views

Why is $T_e \overline{\chi(G)} = \textrm{Im } d \chi$?

Let $G =\textrm{GL}_n$, $s \in G$ diagonalizable, $\sigma: G \rightarrow G$ the automorphism $x \mapsto sxs^{-1}$, and $\chi: G \rightarrow G$ the morphism of varieties $x \mapsto sxs^{-1}x^{-1} = (\...
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1answer
47 views

Exact sequence of sheaves and associated sequence of graded modules

Let $(X,\mathcal{O}_X)$ with $X=\mathbb{P}^n$ and consider a exact sequence of sheaves of $\mathcal{O}_X$-modules $$0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0 $$ Suppose that we apply the ...
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0answers
29 views

Twisting sheaf is invertible.

I have a small question in the proof of Hartshorne's book of the fact that $\mathcal{O}(1)$ is locally free. The thing is that it suffices to prove that $$ \mathcal{O}(1)(D^{+}(f)) \cong \mathcal{O}(...
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0answers
24 views

A Plucker style proof of Monge's Theorem

Plucker, famously, proved Pascal's theorem for all conics at once, using the technique described in the answer here. I was wondering if there was a proof for Monge's Theorem using the above technique....
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2answers
93 views

Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
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1answer
120 views

morphism of sheaves on $\mathbb{R}/\mathbb{Z}$

Let $\mathscr{Z}$ be an arbitrary sheaf on $\mathbb{R}/\mathbb{Z}=X$ (with the quotient topology). Let $\mathscr{F}$ and $\mathscr{G}$ denote the sheaves of continuous functions on $X$ with values in $...
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0answers
33 views

Local ring of an affine curve $K$ at a point $p\in K$

I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows: The local ring of $K$ at $P=(a,b)$ is the ring $$\mathcal{O}_{K,P}=\Gamma(K)_{\mathfrak{m}(a,...
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0answers
39 views

Cartier divisor and map to associated line bundle

Given a Cartier divisor $D$ on an integral, separated scheme $X$ of finite type over an algebrically closed field $k$. Does such a divisor always induce a map $\mathcal{O}_X \to \mathcal O_X(D)$? I ...
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1answer
41 views

Example of an irreducible algebraic set consisting of more than one polynomial

By definition, an algebraic set is a zero locus of polynomials: $$ \{x\in \mathbb A^n \mid p(x) = 0 \,\,\,\, \forall p \in S\}$$ where $S$ is a set of polynomials $p \in k[x_1, \dots, x_n]$. It is ...
5
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1answer
64 views

Ideal of 8 general points in $\mathbb{P}^2$

I am working through chapter 3 of Eisenbud's Geometry of Syzygies. In the first example he makes the claim that the ideal of 8 general points in $\mathbb{P}^2$ is generated by two cubics and a quartic....
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0answers
209 views

How is this fact implicated?

Let $K$ be a field and $\overline{K}$ its algebraic closure, then we define the $n$-dimensional affine space as $$\mathbb{A}^{n}=\{(x_1, \ldots, x_{n})\mid x_1, \ldots, x_{n} \in \overline{K}\}$$ So ...
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0answers
57 views

What is the stalk of the structure sheaf of the plane?

Let $\mathcal{O}$ be the structure sheaf of $\mathbb{A}^2_\mathbb{C}$. How do I compute the local ring corresponding to the stalk of $\mathcal{O}$ of the point $(0,0)$? I tried computing the ...
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1answer
44 views

Isomorphic surfaces in $\mathbb{P}^3$

If $X_0,X_1\subset\mathbb{P}^3$ are surfaces of degree $d\geq 5$ that are isomorphic as abstract surfaces, why is there an automorphism of $\mathbb{P}^3$ that induces an isomorphism between $X_0$ and $...
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0answers
56 views

Which polynomials are resultants?

Let $f(x,y),g(x,y)\in\mathbb{Q}[x,y]$ with degrees $\deg(f)=m,\deg(g)=n$. Considering these polynomials as univariate polynomials in $y$ over the field $\mathbb{Q}[x]$, the resultant $\text{res}(f,g)\...
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1answer
21 views

'Trivial' embeddings have the same degree?

We can define the degree of a projective variety $X\subseteq\mathbb{P}^n$ in terms of the maximal number of intersections with projectivisations $L=\mathbb{P}(\hat{L})$ of linear varieties $\hat{L}\in\...
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2answers
54 views

Hilbert polynomial of iterated Veronese embedding

Let $X=\mathbb{V}(x^2-yz)\subset\mathbb{P}^2$ and consider the Veronese embedding $Y=\mathcal{v}_2(X)\subset\mathbb{P}^5$. Find the Hilbert polynomial, and thus the degree, of $Y$. I know how we can ...
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1answer
44 views

Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
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44 views

Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
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0answers
17 views

Finding Primitive Elements of Separable Function Field Extensions

Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields ...