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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Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
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38 views

Is this ring extension flat?

Let $k$ be a field of characteristic zero and let $A$ be a finitely generated $k$-algebra. Let $B=A[x_1,\ldots,x_n]$ be the polynomial ring over $A$ and let $I \subseteq B$ be an ideal such that $B/I$ ...
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38 views

Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors $$ ...
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20 views

Two questions on surface fibrations

Let $X$ be a smooth surface and $f\colon X\to B$ a fibration, with $B$ a smooth curve. (Q1) Why is the normal bundle of any fiber $F$ trivial? It is clear to me that it has to be of degree zero ...
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1answer
40 views

Understanding the Gluing axiom of the Structure Sheaf on $Spec(R)$

Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category ...
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29 views

Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known ...
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25 views

functor from complex algebraic variety to constructible function

I am reading MacPherson's paper "Chern Classes for singular varieties". Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a ...
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1answer
37 views

The set of polynomials which “cut out” smooth subsets of projective space is open and dense

Let $k[x_0,x_1,...,x_n]_d$ be a space of all forms (in other words, homogenous polynomials) of degree $d$ of variables $x_0, x_1,...,x_n$ over algebraically closed field $k$. Let's think of ...
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22 views

Looking for Severi varieties.

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
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18 views

Intersection of affine subvarieties [on hold]

If the ideals $I_i$ define irreducible subvarieties of an affine space, can the scheme defined by the ideal generated by finitely many of the $I_i$ contain a embedded component?
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21 views

About double couver, linear system and binary quartic forms

I would like to understand the following statement, so I would be happy if someone could help me to understand why it holds. Suppose that $C$ is a nonsingular complete curve of genus 1 over a number ...
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1answer
33 views

cartier duality is a contravariant equivalence?

Can I get a reference for the statement that Cartier duality gives a contravariant equivalence from the category of finite etale group schemes to itself?
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19 views

Domnant morphism

Let $U=U(r,d)$ be the moduli space of stable vector bundles over some curve $X$, and $\Omega$ be the cotangent bundle on $U$. Let $W=\bigoplus _{i=1}^rH^0(X,K_X^i)$. The fiber of $\Omega$ over some ...
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1answer
25 views

involution of affine space fixes a point

Let $\tau$ be a polynomial involution of affine space $k^n$ for $k$ algebraically closed, so $\tau(x_1,...,x_n)=(f_1(x_1,...,x_n),...,f_n(x_1,...,x_n))$ where $f_i$ are polynomials, and ...
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20 views

PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
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44 views

Closed points in projective space correspond to which homogenous prime ideals in $k[x_0,…,x_n]$

I'm trying to think about exercise 4.5.O in Vakil's notes on Algebraic Geometry. Before we defined the scheme $\mathbb{P}^n_k := \operatorname{Proj}(k[x_0,...,x_n])$ and showed that that for $k$ ...
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94 views

Does $\operatorname{Spec}$ preserve pushouts?

The spectrum-functor $$ \operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set} $$ sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a ...
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62 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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1answer
58 views

Image of a line or conic on Veronese surface.

This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid: Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where ...
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20 views

Field of definition of an Ideal

I am trying to prove the following statement from Introduction to commutative algebra and algebraic geometry by Ernst Kunz p,16 Q9 Let $I$ be an ideal of the polynomial ring $K[X_1,X_2,...,X_n]$ over ...
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1answer
26 views

Affine variety over a field which is not algebraically closed

I am now trying to prove the following statement. If the field $K$ is not algebraically closed, then any $K$-variety $V\subset\mathbb{A}$ can be written as the zero set of a single polynomial in ...
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52 views

Proving this fact about algebraic sets.

I want to prove the following equivalence: Let $V$ an algebraic set, $K$ a field and $\overline K$ its algebraic closure. Then we say that $V/K$ ($V$ is defined over $K$) if $I_{V}$ (the ideal ...
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+50

Plücker relations for $Gr_2(\mathbb{C}^5)$

I know that for the complex Grassmannian $Gr_2(\mathbb{C}^4)$, with the index sets $I= \{1,2\}, J = \{3,4\}$, the Plücker relation is given by $$ P_{12}P_{34} - P_{32}P_{14} - P_{13}P_{24} =0 \text{,} ...
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54 views

Fulton , Algebraic Curves, Exercise 2.15

I'm doing some self study on Fulton's Algebraic Curves, and I've done a decent amount. But I'm stuck on a past question that's been bugging me. Question : Let $K$ be a field, and $P,P'$ be points in ...
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35 views

How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point.

How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point. Now $\phi=(F_1/G_1,...,F_n/G_n)$ where $F_i/G_i$ is a regular function. Now how do I conclude?
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1answer
35 views

Projection of the twisted cubic

Question: Let $X$ be the Twisted Cubic in $\mathbb{P}^3$, and $\pi_p:X\rightarrow \mathbb{P}^2$, the projection of the Twisted Cubic from $p$. Find the equations of the projection of the twisted cubic ...
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68 views

Are weakly étale ring homomorphisms of finite presentation étale?

Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat. Question. Are weakly étale ring ...
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1answer
15 views

An example of geometrically integral projective surface that is not smooth

Let $X \subseteq \mathbb{P}^n$ be a geometrically integral projective surface of degree $d$ defined over $\mathbb{Q}$. Does it then imply that $X$ is smooth? I was wondering if someone could provide ...
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30 views

Multiple fibres of an elliptic fibration on Enriques surfaces

Let $X$ be an Enriques surface and $f\colon X\to \Bbb{P}^1$ an elliptic fibration. I will denote by $F$ the general fibre and by $r_1F_1,\cdots,r_kF_k$ the multiple fibres of the fibration. I wanted ...
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31 views

can a field be embedded into product of hyperbolas as a closed subset?

Let's say we have a algebraically closed field $k$. Let $H=\{(x_1,x_2)\in k^2 : x_1x_2=1\}$ be an affine variety. My question is, can $k^1$ be embedded into $H \times H$ as a closed subset? My idea ...
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1answer
33 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
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62 views

Affine varieties and their ideals

I was reading on Wikipedia about quotient ideals. It mentions that if $W$ and $V$ are affine varieties (assume $V$ is) and $I(V)$ and $I(W)$ are the ideals for $V$ and $W$, then $$I(V):I(W) = ...
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1answer
27 views

Self-intersection number in Projective Space

The question is based on the example given in Intersection Theory under the heading Self-intersection. The example is as follows: Consider a line $L$ in the projective plane $\mathbb{CP}^{2}$: it has ...
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How the normal bundle of a divisor changes under a (fnite) quotient map

Let X be smooth and projective, D is an Cartier divisor of X. $\mathcal{N}_{D/X}$ is the normal bundle of D in X. Let $q: X\longrightarrow Y$ be a finite quotient map of degree d. And q is totally ...
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1answer
47 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
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38 views

Why is a projective variety 'the best kind'?

In Hartshorne's AG, he discusses the classification of curves by birational equivalence class says 'based on the idea that a nonsingular projective variety is the best kind..'. What exactly makes a ...
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1answer
28 views

Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
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2answers
66 views

how to prove that an algebraic variety is normal but not smooth?

Let $X=\{(x_1,x_2,x_3) \in \mathbb C^3 : x_1^2=x_2^2+x_3^2\}$, an algebraic variety. How do i prove that $X$ is normal, but not smooth? I guess the non-smoothness appears at the point $(0,0,0)$, but I ...
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1answer
19 views

Infinitely many non-isomorphic degree 8, dimension zero schemes in the plane

In Geometry of Schemes by Eisenbud and Harris, it is claimed in Exercise II-19 that: There are infinitely many isomorphism types of degree 7 subschemes supported at the origin in 3-space, and ...
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1answer
52 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
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24 views

vector bundles of $\mathbb{P}^2$ [on hold]

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
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1answer
45 views

Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
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26 views

Genus of a curve [on hold]

Let $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{D}$ are given. I have tried to find the genus of the following curve, and I have found that it has genus zero in $\mathbb{C}^2$, but if I restrict to ...
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30 views

Nullstellensatz to prove Noether Normalization

In many commutative algebra texts, Noether Normalization Lemma is proved and then Hilbert's Nullstellensatz is obtained as a corollary. Nullstellensatz and Normalization Lemma seem to be non-trivial ...
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27 views

Why the canonical bundle of a complex manifold is a line bundle?

I think I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1. That is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector space ...
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30 views

Is the dual of a flat module flat

Let $k$ be an algebraically closed field, $T$ be a integral, regular, projective $k$-scheme and $X$ another projective, integral $k$-scheme. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k ...
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Distance between point and ellipse - explanation of a paper

EDIT: I notice that the link is hidden, but this post is made with reference to THIS PAPER I'm trying to solve quite an old problem (once again) - to find the distance between a point (in 3d space) ...
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1answer
33 views

Which projective varieties are étale over affine space?

In Liu's answer to this MO question there is a characterization of smooth affine varieties which are étale over affine space. I was wondering if one can give a similar characterization for projective ...
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1answer
30 views

$I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$

How to prove $I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$? Clearly $\sqrt{I(X_1)+I(X_2)} \subseteq I(X_1 \cap X_2)$ But for $f \in I(X_1 \cap X_2)$ $f(x)=0 \forall x\in X_1 \cap X_2$. how to show $f \in ...
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1answer
56 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...