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. ...

learn more… | top users | synonyms (1)

0
votes
0answers
76 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 degrees coming out of a corner and ...
4
votes
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 $X$ ...
5
votes
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 ...
1
vote
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.
6
votes
1answer
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 ...
3
votes
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 ...
12
votes
0answers
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 ...
2
votes
0answers
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. ...
1
vote
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 ...
2
votes
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 ...
3
votes
0answers
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 ...
10
votes
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 ...
4
votes
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 ...
4
votes
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
votes
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?
3
votes
1answer
435 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 $f\in\Gamma(X)$, ...
17
votes
3answers
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 ...
2
votes
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 ...
2
votes
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. $\text{Im } f = \mathcal{G}$. ...
3
votes
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 ...
3
votes
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 ...
7
votes
2answers
401 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 ...
2
votes
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 $k$ ...
4
votes
0answers
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 ...
3
votes
3answers
651 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 ...
1
vote
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
votes
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 ...
2
votes
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 ...
8
votes
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 ...
1
vote
2answers
2k views

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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
4
votes
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
votes
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
votes
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
votes
0answers
175 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 ...
3
votes
2answers
63 views

What is the rationale for the factor of $4$ in the Conics parabola equation?

The Conics form of a parabola equation is $4p(y-k)=(x-h)^2$ where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus. (Which is also the same distance from the ...
7
votes
3answers
188 views

The genus of a curve with a group structure

I'm reading Milne's Elliptic Curves and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1. In this question a similar ...
2
votes
1answer
180 views

Complete intersection

I am trying to solve the following, but I am stuck. Let $P=\mathbb{P}_k^d $ be a projective space over a field k. Let $X$ be a complete subvariety of $P$ of dimension $r$. Say that X is a complete ...
1
vote
1answer
82 views

Some exact sequences of cohomology on picard schemes

I'm looking for the answer of this question. $X$ is a variety over a field $k$, and $Art_k$ is the category of local artinian $k$-algebras whose residue field is $k$. I consider the formal completion ...
5
votes
2answers
187 views

Graphs of transcendental functions are not algebraic varieties

I am trying to show that the zero set of $y - e^x$ is not an affine algebraic variety in $\mathbb{A}^2$. My idea has been to show that any polynomial vanishing on the zeros of $y - e^x$ must vanish on ...
3
votes
0answers
153 views

Algebraic definition of Tangent space of manifold.

Given a smooth manifold $M$ and a point $m\in M$ denote by $\tilde F_m$ the set of germs of smooth functions at $m$. The tangent space at $m$ (denoted by $M_m$)is defined as the space of linear ...
6
votes
1answer
291 views

Notation in Hartshorne Exercise 1.2.6

I am now doing Hartshorne Problem 1.2.6. Hartshorne 1.2.6: Let $Y$ be a projective variety with homogeneous coordinate ring $S(Y)$, show that $\dim S(Y) = \dim Y + 1$. [Hint: Let $\varphi_i ...
8
votes
2answers
149 views

Rank $2$ Elliptic Curves

I'm on a quest for some rank $2$ elliptic curves. My question is actually twofold: Is there a way to easily construct a curve with this property? Is there a database of elliptic curves with given ...
4
votes
1answer
291 views

What are the equations for the image of an algebraically defined subset under the Segre embedding?

Let $\psi: \mathbb{P}^r \times \mathbb{P}^s \to \mathbb{P}^N$ be the Segre embedding with $N = rs + r + s$, as in Hartshorne exercise I.2.14. To be explicit: the image of the pair $([a_0 : \ldots : ...
7
votes
1answer
446 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
10
votes
1answer
886 views

Intuition behind Hilbert's Nullstellensatz

maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof, however I cannot understand the concept in a more ...
3
votes
0answers
270 views

Every smooth cubic curve has a flex point

I want to show that every smooth irreducible plane cubic $C$ has a flex point, i.e. a point $P$ with $i_P(C, T_C(P)) = 3)$ (where $T_C(P)$ is the tangent to $C$ at $P$). I know how to do this in ...