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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4
votes
1answer
52 views

Varieties and ideals

I'm doing the exercises from Fulton of Algebraic Geometry and I'm stuck in the problem 2.44 Let $V$ be a variety in $\mathbb{A}^{n}$, $I=I(V)\subset k[x_{1},\ldots,x_{n}]$, $P\in V$ and let $J$ be ...
3
votes
1answer
139 views

Proving exactness of the conormal sequence

Problem: Let $\phi \colon A \to B$ be a surjective homomorphism of $R$-algebras with kernel $I$. I want to show that the conormal sequence $$ I/I{}^2 \longrightarrow B \otimes_A \Omega_{A/R} ...
0
votes
0answers
29 views

Cohomology of conic bundle 3-folds

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...
15
votes
1answer
143 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
2
votes
0answers
54 views

Motive of a curve and its Jacobian

Let $C$ be a smooth projective curve with a $k-$rational point $x_0$ and $J$ its Jacobian variety. Let us consider the (almost) canonical embedding $j:C \to J$ that sends $x_0$ to the identity $e \in ...
3
votes
2answers
503 views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
2
votes
0answers
28 views

etale morphism between sheaves

We knoe that if $f$ and $ f\circ g$ are both etale morphisms between schemes, then so is $g$. Does this statement hold for etale morphisms between sheavs on etale site over a scheme? More generally, ...
2
votes
1answer
40 views

Dense basic open set contained in dense open subset

For an affine variety $X$ with coordinate ring $A$ it is not hard to see that for $g\in A$ the basic open set (or distinguished open set) $$D(g):=\{ P\in X | g(P)\neq 0\}$$ is dense in $X$ if and only ...
0
votes
0answers
50 views

Any quartic in $\mathbb P^3$ contains only finitely many lines.

I want to prove thath any quartic $X$ in $\mathbb P^3$ contains finitely many lines, but I don't know any method for computing lines on a surface. What is the idea of the proof?
1
vote
0answers
17 views

Amoeba of a line in the plane: An example

Let $z+w+1=0$ a line in $\mathbb{C}^2$ and let $x=log|z| \ge 0$ and $y=log|w|$. I have to show that $$ log(e^x-1) \le y \le 1+e^x $$ But I can't do it! Can you help me, please?
3
votes
0answers
39 views

Normal projective varieties and its coordinate ring

Let $k[X_0,...,X_n]$ be a polynomial ring over an algebraically closed field of characteristic zero and $I$ an ideal of $k[X_0,...,X_n]$ generated by homogenous polynomials. Denote by $X$ the ...
0
votes
1answer
31 views

Question related to dimension of a manifold of zeros over $\mathbb{R}$

Let $F \in \mathbb{z}[x_1,\ldots, x_n]$ be a form of degree $d>1$. Let $V_{\mathbb{R}}$ be the manifold of real zeros of $F$ (I am using the notation of an article I am reading here). Let $B = ...
2
votes
1answer
60 views

Help with Math software (macaulay 2)

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...
3
votes
1answer
44 views

Question about certain morphism between affine spaces

Let the map $\varphi:\mathbb{A}^2\to\mathbb{A}^4$ is given by $$(x,y)\mapsto(x, xy, y(y-1), y^2(y-1)).$$ How to find the system of equations, which defines the image of $\varphi$? If we denote ...
1
vote
1answer
44 views

Exercise about Zariski-topology

I'm trying to became familiar with the basic notions of algebraic geometry and I proved the fact which states every continuous mapping from $\mathbb R^n$ with the Zariski topology to $\mathbb R$ with ...
2
votes
1answer
76 views

How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
0
votes
0answers
14 views

Computing local coordinates

Let $p=[x_0,y_0,1]\in \mathbb P_2(k)$ (projective space). Determine a projective transformation $\phi\in GL(3,k)$ such that $\phi(p)=[0,0,1]$ and name the coordinates explicit. Its easy to see that ...
0
votes
0answers
51 views

Serre's criterion and closure

Let $X$ be a projective scheme of pure dimension $n \ge 2$. Let $U$ be an open dense subset of $X$ such that $\mathrm{codim}(X\backslash U,X) \ge 2$ and for all points $x \in U$, $\mathcal{O}_{X,x}$ ...
1
vote
0answers
42 views

Is exceptional divisor always a Projective bundle over the centre?

Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$ In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the ...
1
vote
1answer
51 views

Ideal sheaf of intersection of two surfaces in $\mathbb P^3$

Let X be an intersection of 2 surfaces of degree $d_1,d_2$ in $\mathbb P^3$. Is it true that there is a short exact sequence $$ ...
-1
votes
0answers
39 views

Easy ways to calculate $\dim \mathbb{K}[x,y]/(f,g)$ [on hold]

Let $f,g \in \mathbb{K}[x,y]$ be to polynomials without common divisors and $\deg f = n, \deg g = m$. I want to prove that $\dim\mathbb{K}[x,y]/(f,g) \leq n\cdot m$. I know only that $|V((f,g))| \leq ...
0
votes
0answers
91 views

Notation on De Jong and Starr's paper

I reading De Jong and Starr's paper "Divisor classes and the virtual canonical bundle for genus 0 maps". I am quite confused about what would be the correct definition of functor ...
1
vote
0answers
70 views

A smooth rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique smooth quadric surface

I am trying to solve the exercise IV.6.1 from Hartshorne's "Algebraic geometry": A smooth rational curve of degree 4 in $\mathbb{P}^3$ is contained in a unique quadric surface $Q$, and $Q$ is ...
7
votes
0answers
164 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{2}$

Let $f,g:\mathbb{C}^3\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^3\to \mathbb{C}^3$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
3
votes
0answers
84 views

are connnected components of this scheme irreducible?

So I have a normal surface (ie, all components are dimension 2) $X$ which is smooth and affine over a Dedekind domain $R$ (so $X$ is an affine scheme). Suppose $R'$ is integral (possibly not dedekind) ...
1
vote
1answer
30 views

For $0\to F\to E\stackrel{\varphi}{\to} L\to 0$, why is the pullback under $\varphi$ of a constant section an $F$-torsor?

I think the following is used in classifying $F$-torsors. Let's take $0\to F\to E\stackrel{\varphi}{\to} L\to 0$ to be an exact sequence of vector bundles, all over a scheme $S$, where ...
16
votes
4answers
2k views

“Real”-life applications of algebraic geometry

Before you tell me that this question has been asked, give me a bit of your time please to read this question because it is not as simple as it sounds. I did my undergraduate degree in mathematics, ...
5
votes
2answers
70 views

De Rham-Weil theorem

I am having trouble understanding a couple of points with regard to the De Rham-Weil theorem and was hoping that someone might be able to shed some light. Let $X$ be a smooth manifold and ...
0
votes
1answer
44 views

Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
4
votes
1answer
89 views

Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
3
votes
1answer
43 views

Algebraic Curves and Second Order Differential Equations

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve. I am aware that the Weierstrass $\wp$ - ...
0
votes
1answer
41 views

Is the closure of a set $A$ in the Zariski topology $V(A)$?

Let $R$ be a commutative unitary ring and $A\subseteq \operatorname{Spec}(R)$ a subset. If $A=\{p\}$, then the closure $\bar A$ of $A$ equals $V(p)=\{x\in \operatorname{Spec}(R) \mid p\subseteq x\}$. ...
2
votes
2answers
67 views

Duality and Serre's criterion

Let $X$ be a projective scheme, $\mathcal{F}$ a coherent sheaf on $X$ which is $S_2$. Then under what additional conditions is its dual, $\mathcal{H}om_X(\mathcal{F},\mathcal{O}_X)$ also $S_2$?
1
vote
1answer
53 views

Group theory and Complex Analysis

Actually this time I am having interest in group Theory and complex analysis. So I want to study the topics which relate both. So please suggest me which topic or book should I read to explore the ...
1
vote
1answer
32 views

find equally spaced points on parabola

I'm trying to find equally spaced points on a parabola simply defined by $$y = \frac{x^2}{2 p}$$ Someone told me there is an easy way to split the parabola but he didn't tell me how and I cannot find ...
2
votes
1answer
43 views

A question about endomorphism rings of elliptic curves

This is probably a very trivial question, but I haven't been able to find a rigorous explanation anywhere so far or at least haven't understood it. Assume we have an elliptic curve $E$ over ...
3
votes
0answers
45 views

Confusion over common criterion for a line bundle on a scheme to be trivial?

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to ...
3
votes
2answers
71 views

Prerequisite of Algebraic Geometry

Algebraic geometry, as far as I know, is a very important branch of mathematics, which is also very difficult. I am going to take a try to taste that. Before really going into the field, I have two ...
3
votes
1answer
64 views

What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course ...
1
vote
2answers
60 views

Sheafification, stalks and quotient

I gave a problem that I can't finish by myself. Any help would be appreciated. Consider a sheaf $\mathcal{F}$ of abelian groups on a topological space. I would like to show that given two sheaves ...
2
votes
0answers
27 views

Inverse limit of irreducible spaces

Let $(X_{i})_{i \in \mathbb{N}}$ be an inverse system of topological spaces. Assume that each of the $X_{i}$ is irreducible. Then is it true that $\projlim X_{i}$ is also irreducible? I read in a ...
1
vote
0answers
36 views

are extensions of flat connections flat?

Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a ...
3
votes
1answer
36 views

Question about characteristic polynomial of the Frobenius endomorphism on elliptic curves.

I have another possibly trivial question about elliptic curves. A lot of papers I've seen state that the characteristic polynomial of the Frobenius endomorphism of an elliptic curve over a finite ...
1
vote
1answer
39 views

what is the precise definition of a morphism defined over $k$?

Let $k$ be a field and $X$ an algebraic variety over $k$. I have often seen people write $Aut(X/k)$ for the automorphisms "defined over k". What is the exact definition? My guess is that $X$ comes ...
0
votes
2answers
46 views

Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
2
votes
1answer
31 views

Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings?

A brief question, if $\pi\colon E\to X$ is a vector bundle over a scheme $X$, is it automatic that any section $s\colon X\to E$ always a closed embedding?
0
votes
1answer
34 views

Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings?

Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings? Is it of finite type? Is the map $\operatorname{Spec} \prod_{i=1}^\infty \mathbb{C} ...
1
vote
0answers
18 views

Double Coset Space

In the theory of Shimura varieties you construct the variety as a double coset space $G(\mathbb{Q}) \backslash X \times G(\mathbb{A}_f) / K$ . I do not understand this as a variety. I thought you ...
1
vote
2answers
50 views

What's wrong with my example?

I have been asked to show that if $V\subset k^n$ is an affine algebraic variety over an algebraically closed field $k$, and $dom(f) = V$ for some $f\in k(V)$ then $f$ lies in $k[V]$. Here, $k(V)$ is ...
1
vote
0answers
35 views

dimension of the span of all partial derivatives of a given polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...