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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Rational projective variety given by coprime homogeneous polynomials

Let $K$ be algebraically closed field. Let $f_k, f_{k-1} \in K[x_1,...,x_n]$ coprime homogeneous polynomial of degree $k$ and $k-1$ respectively. I want to prove that: The variety $$ X = ...
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2answers
615 views

Exercise 2.4 Fulton's Algebraic Curves

I am looking at exercise 2.4 in William Fulton's "Algebraic Curves". It asks to prove that if $X\subset \mathbb{A}^n$ is nonempty affine variety, then the following are equivalent $X$ is a point ...
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1answer
97 views

Planes, quadric surfaces and then …?

If planes are described as: $\mathbf{n} \cdot (\mathbf{r}-\mathbf{r_0})=0$ And quadric surfaces can be described as: $\mathbf{x}^T\mathbf{A}\mathbf{x} = 0$ (with $\mathbf{x} = \begin{bmatrix} x ...
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0answers
65 views

Length of a quotient in Hartshorne, IV, Ex. 1.8.

In Hartshorne's Algebraic Geometry, IV, Ex. 1.8., he considers $\text{length}(\tilde{\mathscr{O}_P}/\mathscr{O}_P)$, where $\mathscr{O}_P$ is the local ring of $X$ (considered also as the skyscraper ...
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1answer
186 views

Definition of prime ideal sheaf

In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf": Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let ...
6
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1answer
123 views

Showing that $\mathbb{C}[x,y]^{\mu_n}$ and $\mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic as rings

The problem: Let $\mu_n$ act on $\mathbb{C}[u,v]$ with weights $(1,-1)$. I would like to show that the rings $\mathbb{C}[u,v]^{\mu_n}$ and $\mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic. Explanation of ...
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2answers
560 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
3
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0answers
98 views

A finite type morphism between regular schemes and closed immersions.

I am working through the following problem in Qing Liu's book on Algebraic Geometry, 6.2.6 a), which reads: Let $X \rightarrow Y$ be a morphism of finite type of locally noetherian regular ...
2
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1answer
74 views

Extension of prime ideal in $k[V]$ to $\mathcal{O}_P(V)$ is prime?

Let $k$ be an algebraically closed field, $I\subset k[X_1,\cdots, X_n]$ be a prime ideal, $V=V(I) \subset \mathbb{A}^n$ a variety and $P=(a_1,\cdots, a_n)\in V.$ Recall that $\mathcal{O}_P(V)$ is the ...
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1answer
145 views

Inverse image of Veronese like map

The following is a very elementary question but I can't find the error: Denote $D$ the diagonal $\{[x_0:x_1],[x_0:x_1]\} \subset \mathbb P^1$x $\mathbb P^1$. Let $\phi: \mathbb P^1 \longrightarrow ...
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0answers
55 views

Algorithm for checking that a singularity is an ordinary double point

Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically ...
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2answers
282 views

What does the stalk of a sheaf of discontinuous sections look like?

I'm having some kind of cognitive dissonance here, but I'm having trouble figuring out which of my beliefs is false. Let $\mathscr{F}$ be a sheaf of abelian groups on a topological space $X$ and $x ...
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0answers
84 views

Geometry Question with irregular hexagons

Suppose you have a rectangle with sides $x$ and $y$ and both numbers are integers and have no factors. now draw lines inside this rectangle starting with a line at $45^\circ$ coming out of a corner ...
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1answer
272 views

Pullback of sheaves and pullback of schemes

Let $\mathbb{G}_m$ the multiplicative group, with coordinate ring $\mathbb{C}[x^{\pm 1}]$, and considered as a sheaf of abelian groups over $\mathrm{Spec}\,\mathbb{C}$ in the Zariski topology. Let $X$ ...
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1answer
196 views

Vakil 14.2.E: $L\approx O_X(div(s))$ for s a rational section.

I am working through Vakil's Ch 14 (march2313 version) on invertible sheaves and am having trouble on 14.2.E. The question (in notation to be defined) is this: how do I show that each point in the ...
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1answer
53 views

Finding a presentation of $A$-algebra $B$

Find a presentation of the $A$-algebra $B$, where $B=\mathbb{Z}[1/2]\subseteq \mathbb{Q}$ and $A= \mathbb{Z}$. I want to prove it but I can't understand what want to me! Please describe to me.
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76 views

Function field question from Silverman's AEC

Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring. Then he states Proposition 1.7 (the intrinsic characterization of ...
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1answer
99 views

Computing a rational function at a point in terms of a uniformising parameter

I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
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501 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
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72 views

Uniqueness of minimal resolution

Let $R$ be a domain, and $a_1,\dots,a_r$ be a regular sequence of $R$. Let $b_1,\dots,b_r$ be another regular sequence, such that two regular sequences generate the same ideal, i.e. ...
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1answer
133 views

The set of rational points of a scheme of finite type over a field.

In this book http://ukcatalogue.oup.com/product/9780199202492.do#.UYDnvZNk1bA (Liu's Algebraic Geometry book), we can find the next proposition; Proposition 3.2.20. Let $X$ be a geometrically reduced ...
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1answer
117 views

Cremona transformation and line arrangements

We work over the complex numbers. Let $A_3 \subseteq \mathbb{P}^2 $ be the following arrangement: take three generic lines in the plane and pass an ellipse through the three intersection points. We ...
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428 views

On the definition of a normal crossing divisor

I'm reading a material that states: Definition: Let F be a foliation on a analytical manifold N. A normal crossing divisor on N is a collection of submanifolds $E$ of $N$ such that for every point ...
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1answer
286 views

Intersection of powers of maximal ideals

Let $A=\mathbb K[X_1,\ldots,X_n]$ be a polynomial ring over some field $\mathbb K$. Let $\mathfrak p\subseteq A$ be a prime ideal. Let $Z(\mathfrak p)=\{ \mathfrak m\subset A\text{ maximal}\mid ...
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2answers
167 views

Hartshorne Ex. 1.3.8 - Where do I take intersections here?

Let $H_i$ and $H_j$ be the hyperplanes in $\Bbb{P}^n$ defined by $x_i = 0$ and $x_j = 0$ with $i \neq j$. I want to show that any regular function on $\Bbb{P}^n - (H_i \cap H_j)$ is constant. Now I ...
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1answer
423 views

A question on torsion sheaves

Im not sure if Ive got this right: Let X be an integral scheme and $\mathcal{F}$ a coherent sheaf. Then $\mathcal{F}$ is torsion if and only if it is not supported at the generic point. It is is easy ...
13
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1answer
310 views

$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
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1answer
438 views

A principal open set of an affine algebraic set is an affine variety

Notations $k$ is an algebraic closed field and $\mathbb A^n(k)$ is the topological space $k^n$ with the Zariski topology If $X\subseteq\mathbb A^n(k)$ is an affine algebraic set and $f\in\Gamma(X)$, ...
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789 views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
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1answer
371 views

Questions on Reduced Induced Closed Subscheme

I've just read the definition of a closed subscheme in Hartshorne's recently and I collected here and there (notes that people put online) the following statement. Claim. Suppose that ...
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2answers
112 views

A Criterion for Surjectivity of Morphisms of Sheaves?

Suppose that $f: \mathcal{F} \rightarrow \mathcal{G}$ is a morphism of sheaves on a topological space $X$. Consider the following statements. 1) $f$ is surjective, i.e. $\text{Im } f = \mathcal{G}$. ...
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1answer
117 views

Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
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1answer
121 views

Vector space structure on $\mathcal{O}/\mathfrak{m^n}$

Let $k$ be a field and $F\in k[X,Y]$ irreducible such that $F(0,0)=0$. Let $\mathcal{O}$ the local ring of the plane curve $F$ at $P=(0,0)$ and suppose that $P$ is a simple point of $F$. Suppose that ...
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2answers
402 views

Principal maximal ideals in coordinate ring of an elliptic curve

Let $E$ be an elliptic curve over an algebraically closed field, and let $R$ be the coordinate ring of $E \setminus \{\infty\}$. I have read somewhere that $R$ has no principal maximal ideal. But I ...
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1answer
75 views

Bijection between hom sets of $k$ - algebras

Let $R:= k[x_1,\ldots,x_r]$, $S:= k[x_{r+1},\ldots,x_{r+s}]$ and $Q:= k[x_1,\ldots,x_{r+s}]$. Let $I \subseteq R$ and $J \subseteq S$ be ideals. I have in texts in algebraic geometry that for any $k$ ...
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162 views

Inverse images under universally injective morphisms

Let Y be locally Noetherian, and consider a projective morphism $f:X \rightarrow Y$ such that the map $\textbf{Spec} f_\ast \mathcal{O}_X \rightarrow Y $ is universally injective. Let $C \rightarrow ...
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3answers
652 views

Ring of Regular functions on $\Bbb{A}^2 - \{(0,0)\}$

Suppose I want to determine the ring of regular functions on $U = \Bbb{A}^2 - \{(0,0)\}$. Now I can do this assuming the following fact: Fact: If $f$ is regular on $U$, then we can write $f ...
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1answer
65 views

Proposition 3 in Chapter I.7 (Dimension) of Mumford's Red Book

In Mumford's Red book, chapter I.7 (Dimension), the proof of Proposition 3 (1.) has the step: If $B=f^{\star -1}(A)$, apply the going-up theorem to $S/B\subset R/A$. What does the inclusion ...
6
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1answer
42 views

Showing the $V(\mathfrak{a})$ give us a topology on Proj$S$

I'm a bit confused about the proof of Lemma 2.4 on page 76 of Hartshorne's Algebraic Geometry: Lemma 2.4 (a) If $\mathfrak{a}$ and $\mathfrak{b}$ are homogeneous ideals in $S$, then ...
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0answers
97 views

Induced Sheaf on Subspaces

Suppose that $X$ is a topological space, $\mathcal{F}$ is a sheaf (of abelian groups, rings, ideals, modules) on $X$ and $Y \subset X$. Do you know a natural way to get an induced sheaf on $Y$ from ...
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1answer
212 views

Intersection of Algebraic Varieties.

Let $\mathbb K$ be an algebraically closed field. Consider the set $M_n(\mathbb K)$ of all matrices of order $n$. Identify the set $M_n(\mathbb K)$ with the affine space $\mathbb A^{n^2}_{\mathbb ...
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Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$

There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents ...
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0answers
84 views

Exercise B.5.(c) of Hindry and Silverman's Diophantine Geometry

I am trying to do the following exercise: Let $V$ be the projective line and $D$ is the point at infinity. Suppose $f,g$ are morphisms from $V$ to $V$ with degree greater than $1$. Show that $f$ and ...
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1answer
65 views

Pullback of a reduced closed subscheme along an étale morphism

Let $X$ any $Y$ be reduced schemes of finite type over a field and $W\subseteq Y$ a closed reduced subscheme. What is an example of such $X$ and $Y$ and $W$ and of a morphism $f:X\to Y$ such that ...
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1answer
64 views

Lie bracket in local coordinates.

$\bf 14.9.$ Lie bracket in local coordinates Consider the two vector fields $X,Y$ on $\mathbb{R}^n$: $$X=\sum a^i\dfrac\partial{\partial x^i},\qquad Y=\sum b^j\dfrac\partial{\partial x^j},$$ where ...
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3answers
204 views

The analytic and the algebraic “small disc”

I would like to understand the relation between an analytic object (the so called "small disc") and an algebraic one (the spectrum of a DVR). The framework is that of one-parameter families of complex ...
2
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1answer
196 views

criterion for geometrically integral scheme

I want to prove the remark 3.2.9 of the book Algebraic Geometry of Arithmetic Curves (of Quing Liu) that is: let $X$ be an integral scheme with function field $K(X)$, if $K(X)\otimes_k \overline{k}$ ...
5
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0answers
102 views

Why is an intersection product $X.C=0$?

Here is the situation: $S$ is a nonsingular complex projective surface, and $C=X\cup_AY\subset S$ is a uninodal curve of compact type: it is obtained by glueing two nonsingular curves $X,Y\subset S$ ...
3
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0answers
176 views

Finite type ring extension + condition = finite extension?

Is the following true ? If $A \subset B$ is finite type extension (i.e. $B$ is a finitely generated $A$-algebra) of integral domains such that the set $\{\mathfrak ...
3
votes
0answers
381 views

Is this incidence variety in $\mathbb{P}^2 \times \mathbb{P}^2$ isomorphic to a variety in $\mathbb{P}^1 \times \mathbb{P}^1$?

I have an incidence variety $X = \{(p,\ell) \in C \times D^* : p \in \ell\} \subset \mathbb{P}^2 \times \mathbb{P}^2$, where $C = Z(f) \subset \mathbb{P}^2$ and $D^* = Z(g^*) \subset \mathbb{P}^2$ are ...