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)

2
votes
1answer
48 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 ...
1
vote
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. ...
0
votes
0answers
37 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 ...
0
votes
0answers
27 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 ...
0
votes
0answers
36 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 ...
2
votes
0answers
49 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 ...
1
vote
1answer
62 views

the definition of “Birational Equivalence”

I have confused with the definition of "Birational Equivalence" in the Algebraic Geometry. In My Text book , ($V$ and $W$ are irreducible quasi-projective varieties) A rational map $f : V \to W$ is ...
1
vote
1answer
55 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 ...
2
votes
1answer
47 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 ...
-1
votes
1answer
31 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 ...
1
vote
0answers
32 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 ...
3
votes
2answers
229 views

Kernel of a morphism from a locally free sheaf is locally free

Let $C$ be a projective curve (not necessarily reduced or irreducible). Let $\mathcal{F}, \mathcal{G}$ be $\mathcal{O}_C$-modules and $\phi:\mathcal{F} \to \mathcal{G}$ be a morphism of ...
0
votes
1answer
26 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 ...
3
votes
2answers
47 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 ...
0
votes
0answers
22 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 ...
1
vote
0answers
22 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 ...
0
votes
1answer
93 views

Plotting 3 equidistant points on a sphere

I'm trying to figure out how to plot with $x,y,z$, three points that are equidistant along the surface of a sphere from each other that are all on a horizontal axis (so $y = 0$) with a radius of $500$ ...
2
votes
1answer
39 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 $1$?
0
votes
1answer
67 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point ...
5
votes
1answer
76 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 ...
1
vote
1answer
69 views

Finiteness of Zeros and Poles on Noetherian schemes

This exercise comes from Ravi Vakil's notes. Suppose that $X$ is an integral Noetherian scheme, and $f \in K(X)^{\times }$ is a nonzero element of its function field. Show that $f$ has a finite number ...
1
vote
1answer
43 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 ...
5
votes
2answers
124 views

Why are projective spaces and varieties preferable?

I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why. In a more elementary stage of mathematics, when we try to find ...
10
votes
1answer
740 views

What conditions guarantee that all maximal ideals have the same height?

It fails in general that all maximal ideals in a commutative ring with unity have the same height. It's easy to construct a counter-example when the ring is NOT an integral domain (consider the ...
0
votes
1answer
17 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} = ...
1
vote
0answers
28 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 ...
0
votes
0answers
23 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 ...
1
vote
0answers
77 views

Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...
1
vote
0answers
29 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 ...
1
vote
2answers
51 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 ...
2
votes
4answers
1k views

Intersection of ellipse with circle

I would like know whether a circle is intersecting an ellipse. Here ellipse equation is $$Ax^2 + Bxy + Cy^2 + dx+ey + 1 = 0,$$ and the circle equation is $$(x-g)^2 + (y-f)^2= r^2.$$
0
votes
0answers
37 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 ...
0
votes
1answer
38 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
votes
1answer
59 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 ...
0
votes
0answers
46 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 ...
1
vote
0answers
37 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 ...
1
vote
1answer
42 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 ...
1
vote
1answer
20 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 ...
1
vote
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 ...
1
vote
1answer
29 views

Action of the group of automorphisms of a connected finite étale cover (Corollary 5.3.4 in Szamuely).

I have difficulty to understand the proof of Corollary 5.3.4 in Szamuely, Galois group and fundamental groups. The proof use the following corollary. Corollary 5.3.3. If $Z \longrightarrow S$ is a ...
4
votes
0answers
86 views

Understanding the topology of $y^2=(x-1)(x-2)(x-3)(x-4)$

Andreas Gathmann's lecture notes on algebraic geometry start by considering the curve $C_n=\{(x,y): y^2 = (x-1)(x-2)...(x-2n)\} \subset \mathbb{C}^2$. He claims that the topology of this curve is the ...
1
vote
0answers
43 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 ...
0
votes
0answers
12 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 ...
3
votes
0answers
39 views

Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
4
votes
1answer
400 views

Extension of regular function

This is an exercise in Hartshorne's book. For a quasi projective variety $Y$ with dimension $\geq 2$ and $p \in Y$ a normal point, if $f$ is regular on $Y-\{p\}$ then $f$ can be extended to a ...
4
votes
1answer
68 views

History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they ...
0
votes
1answer
62 views

About alternative ways of computing $H^1(X,\mathcal{O}_{\mathbb{P}^n}(m))$.

This is a follow up to my question : Applications of $Ext^n$ in algebraic geometry In the case of $\mathcal{O}_X$-Modules it is clear that $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ ...
4
votes
1answer
86 views

About birational map between curves of the form $y^{2}=g(x)$

I'm trying to solve the following exercise but I'm stuck after trying for a long time. Suppose that $g(x)=ax^{4}+bx^{3}+cx^{2}+dx+e\in{k[x]}$ and similarly ...
18
votes
2answers
2k views

So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
0
votes
1answer
30 views

Local argument for proving that $\operatorname{Ext}^1$ vanishes for two sheaves

I am trying to compute the vanishing of $\operatorname{Ext}^1$ for two sheaves of $\mathcal{O}_X$-Modules and I was wondering if it was possible to use some local argument to reduce the problem to ...