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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3
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0answers
36 views

Find a subsheaf of a coherent sheaf, whose quotient is torsion sheaf.

Suppose $X$ is a variety, $\mathcal{F}$ is a coherent sheaf, its generic stalk has rank $r$. How to find an ideal sheaf $I$ such that a direct sum of $r$ copies of $I$ injects into $\mathcal{F}$ and ...
0
votes
1answer
37 views

Equidistant points on a circle

I would like to obtain/generate points on a circle in Cartesian coordinates such that the distance between two consecutive points will be always equal. For example, plotting a circle with radius 100 ...
0
votes
1answer
35 views

Help in this easy equivalence

If $C$ is a curve with genus $g$ and $k$ a field, I'm stuck in something I'm sure easy, I think I'm forgetting some basic things. Define $\Omega(D)=\{\omega\in\Omega;div(\omega)\ge D\}$ and ...
3
votes
1answer
90 views

Except First year Abstract Algebra and commutative Algebra, what else do i need to start read Algebraic Geometry text?

Except First year Abstract Algebra and commutative Algebra text, what else do i need to read before start read Algebraic Geometry texts? I am refer to the beginning texts: "Algebraic geometry an ...
3
votes
0answers
66 views

Leray-Hirsch Using Kunneth Formula from “Differential form in Algebraic Topology” by Bott and Tu

Kunneth Formula: Let M and F are manifolds. If M has a finite good cover then $H^n(M\times F)=\bigoplus _{p+q=n} H^p(M)\bigotimes H^q (F)$ Bott and tu says One can prove Leray-Hirsch theorem by the ...
11
votes
2answers
482 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 ...
10
votes
2answers
304 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
0
votes
1answer
36 views

How to find certain quadratic curves over $\mathbb{Q}$

Given a quartic curve C: $x^4+y^4=1$, how can I find a quadratic curve over $\mathbb{Q}$ intersecting $C$ at four points, while the intersection multiplicity of each point is 2?
2
votes
2answers
46 views

(Reference Request) Desingularization of Fibrations

I am interested in morphisms from a projective surface to a projective curve $p: X \longrightarrow C$ such that the fibres are singular curves on $X$. In the best case scenario, I'd like to replace ...
-1
votes
1answer
31 views

Show the curve has no factors degree 1 or 2

I have to show that $ g(x,y)=(x^{2}−1)^{2}+(y^{2}−1)^{2} $ (over the real numbers) has no factors of degree 1 or 2. I already know that g only consists of the four points (±1,±1) and those are all ...
2
votes
1answer
22 views

$l(rP)\le l((r-1)P)+1$

If $C$ is a curve of genus $g$, I'm trying to prove the dimension of the divisor $rP$ associated to this curve is less or equal than the dimension of the divisor $(r-1)P+1$, where $r\in \mathbb N$. ...
10
votes
0answers
86 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
0
votes
0answers
25 views

Show a curve has no factor of degree 1 or 2

I have to show that $ h(x,y)=y^{2}(x^{2}+x+1)-x^{2} $ has no factors of degree 1 or 2. I know that h contains infinitely many points and is singular at the points (1,0,0), (0,1,0) and (0,0,1). I am ...
1
vote
0answers
39 views

Is $\mathbb{A}^1$ isomorphic to one of its quasi-affine subsets?

It's well known that $\mathbb{A}^1$ is not isomorphic to any proper open subset of itself. Just out of curiosity, is $\mathbb{A}^1$ isomorphic to any proper quasi-affine subset of itself, or is this ...
0
votes
0answers
32 views

Curve has no factor of degree 1 or 2

I have to show that $g(x,y)=(x^2-1)^2+(y^2-1)^2$ (over the real numbers) has no factors of degree 1 or 2. I already know that $g$ only consists of the four points $(\pm 1, \pm 1)$ and those are all ...
2
votes
2answers
79 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 ...
0
votes
1answer
29 views

When is an algebraic curve on the plane a Jordan curve?

It is very intuitive that the set $$ S=\{{x,y}\in\Bbb{R}^2\mid 2x^6+3y^4=1\} $$ is a simple closed curve. How can one show that this is indeed true? Does this question relate to some theorems ...
3
votes
1answer
27 views

Why this dimension is $0$ using Riemann-Roch theorem?

If $C$ is a curve of genus $g$, I'm trying to prove the dimension of the divisor $(2g-1)P$ associated to this curve is $g$. I'm using the Riemann-Roch theorem which says: Let $W$ be a canonical ...
3
votes
0answers
27 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
2
votes
2answers
26 views

If we delete two points $x,y$ from $\mathbb{A}^1$, can we without loss of generality assume $x=0, y=1$?

My intuition is that we can assume this. More precisely, what I mean is, suppose $\mathbb{A}^1_k$ is the affine space over an algebraically closed field $k$. If $x,y$ are any two distinct points in ...
1
vote
1answer
32 views

Polynomial representation of intersection of polynomials

How to minimally represent intersection of two degree $d$ polynomials intersecting at $d^2$ points as a single polynomial?
4
votes
0answers
46 views

Hodge Bundles on Tropical Spaces

I am not sure that this question even makes sense, which I suppose is part of the questions itself. In any case, I attended a talk recently wherin there was some discussion about a "tropical ...
1
vote
0answers
47 views

Coherent sheaf on reduced scheme is free on dense open set

It should be well-known fact, but I couldn't find this in Hartshorne's "Algebraic Geometry", Mumford-Oda or Ravi Vakil's Lecture notes. Let $X$ be a reduced connected scheme and $\mathcal F$ is a ...
2
votes
0answers
33 views

Why does this map is well-defined?

I didn't understand this proof from Fulton's Algebraic curves book: Why $ord_P(f)\ge -r-1$ in order to this map be well-defined? Thanks
1
vote
0answers
19 views

Why does this construction give a proper curve?

Let $k$ be algebraically closed. The claims is there is a functor $\{$ Finitely generated extensions of $k$ of transcendence degree $1$ $\} \rightarrow \{$ Smooth, connected, proper, integral curves ...
2
votes
1answer
44 views

Hartshorne Corollary 9.4

I am reading the section of Flatness from Hartshorne. I have a doubt in the proof of the corollary of the following proposition : $\textbf{Proposition 9.3}$ Let $f:X\longrightarrow Y$ be a separated ...
5
votes
2answers
85 views

Can the Kahler differentials of a “good” local ring R be free of rank not equal to dim(R)?

Let $R$ be a local ring containing a field isomorphic to its residue field $k$. Assume $R$ is a localization of a finitely-generated $k$-algebra. Can $\Omega_{R/k}$ be free of rank $r\neq\dim{R}$? ...
6
votes
1answer
104 views

When does a smooth projective variety X have a free Grothendieck group

Let $X$ be a smooth projective variety (e.g. Grassmannians). Since $X$ is smooth, the groups $G_0(X):=K_0(CohX)$ and $K_0(X):=K_0(VectX)$, the Grothendieck groups of coherent sheaves of modules on $X$ ...
0
votes
0answers
67 views

Induced Spec map for a morphism of finitely generated $\mathbb{C}$-algebras

I have a morphism $f:A\longrightarrow B$ of finitely generated $\mathbb{C}$-algebras. I have proven, using Zariski's lemma, that the inverse image of a maximal ideal $M \subset B$ is a maximal ideal ...
1
vote
0answers
42 views

Splitting of short exact sequence of sheaves

Let $X$ be a smooth projective variety over a field, say $k$. Consider the short exact sequence of $k$-modules, $$0 \to A_1 \to A \to A_2 \to 0$$ where $A$ and $A_2$ are $k$-algebras. Since these can ...
3
votes
0answers
16 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
7
votes
1answer
58 views

Projective morphism defined by linear systems

Let $X$ be a normal variety and $D$ be a Cartier divisor, suppose $\sigma, \delta$ are two basepoint free linear systems in $|D|$, then we have two morphisms defined by these two linear system: ...
0
votes
1answer
29 views

Finite maps between algebraic varieties are closed

In the proof given by the book of Shafarevich "Basic algebraic geometry 1", section 5.3 of chapter 1, that every finite map between algebraic varieties $f:X \longrightarrow Y$ (where we assume $f(X)$ ...
3
votes
0answers
53 views

image of the abel-jacobi map from a hyperelliptic curve

For a fixed point $x_0\in X$ of a hyperelliptic curve(genus $g$), we can think of the image of Abel-Jacobi map $u: x\mapsto (\int_{x_0}^{x}\omega_1,\ldots,\int_{x_0}^{x}\omega_g)$ into its Jacobian ...
2
votes
1answer
37 views

About non ruled surfaces

I have this statement about a surface $S$ (complex projective algebraic variety of complex dimension 2). Take $S$ a non ruled surface. Then the euler characteristic of $S$ satisfies this inequality: ...
2
votes
0answers
18 views

Two definitions of “affine stratification”

I see two different definitions of "affine stratification" in the literature: A stratification where each stratum is isomorphic to $\mathbb{A}^n$ for some $n$. A stratification where each stratum is ...
3
votes
1answer
50 views

Irreducible polynomial and the zero set of its derivative

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Consider the derivative of $P,$ $D_{\mathbb{C}}P$, as a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^n.$ I have the following question: ...
7
votes
1answer
66 views

Three meanings of étale sheaf on X

When I am studying stacky stuffs, I am always confused by the notion of étale abelian sheaves on $X$, because conceivably there might be three different meanings of that: Take the global étale site ...
2
votes
1answer
119 views

Why this is equivalent to proper?

maybe that's an idiot question. Anyway, when I was reading this master thesis http://sma.epfl.ch/~werndli/scripts/mscthesis/elementary_gaga.pdf I could not understand the following equivalence (in ...
1
vote
0answers
52 views

Pre- and post-multiplication by diagonal matrices

Let $\mathbf{1}$ denote an $n\times 1$ vector with all entries equal to 1. Given an $n\times n$ matrix A with strictly positive entries, and non-negative diagonal matrices $D_1$ and $D_2,$ evaluate ...
2
votes
1answer
50 views

Roadmap to Riemann hypothesis for curves over finite fields

I am a beginning graduate student with (almost) no background in algebraic geometry. I would like to learn the proof of the Riemann hypothesis for curves over finite fields, including all ...
0
votes
0answers
39 views

Vector bundles on projective varieties

For instance, in algebraic category, given a vector bundle on a smooth projective variety (irreducible), then is it always true that such bundle can be embedded as a subbundle of a trivial vector ...
1
vote
0answers
35 views

deformation space inside cohomology

For which smooth projective varieties $X$ is $H^1(X,T_X)$ (canonically ) contained in $H^\cdot(X,\mathbb C)$? If $K_X$ is trivial this is true. But are there other type of varieties?
0
votes
1answer
55 views

Help to translate this theorem to a more accessible language

I'm trying to understand the chapter 2 of this article. I'm stuck in this part: The theorem he mentioned is from this book and it is the following: I need help to translate this theorem to a ...
0
votes
1answer
31 views

How to exhibit an étale cover of a surfaces.

I have this statement about complex algebraic projective surfaces. Let $S$ a surface such that the euler characteristic of $S$ is negative. $X_{top}(S)<0$. Then there exist an étale cover $S^{'} ...
0
votes
0answers
19 views

Hilbert series computation for Hilbert scheme of $n$ points on $\mathbb C^2$

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{character}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $T$ acts on $x,y$ as $(t_1,t_2)$? ...
0
votes
1answer
51 views

What is an étale covering

I'm reading Buouville's book: Complex Algebraic Surfaces and in particular i'm interested about the part exposing nice properties of surfaces of general type. During the lecture i'have found the term ...
0
votes
0answers
49 views

Hartshorne's Exercise II. 2.15 (fully faithful functor)

I'm struggling with the last part of the exercise. Namely, let $V,W$ be any two varieties over a field $k$. We build the functor $t$, which induces a natural map $$ ...
1
vote
0answers
41 views

Reference request on numerical semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
1
vote
1answer
89 views

Hartshorne Proposition 9.5

The Proposition is that : Let $f:X\longrightarrow Y$ be a flat morphism of schemes of finite type over a field $k$. For any point $x\in X$, let $y=f(x)$. Then $\dim_x(X_y)=\dim_x(X)-\dim_y(Y)$. Here ...