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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6
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0answers
105 views

Higher direct image of morphism with generic fiber $\mathbb{P}^1$

Let $f:X\to Y$ be the morphism of smooth varieties over $\mathbb{C}$ with generic fiber equal to $\mathbb{P}^1$. How to prove that $R^if_*\mathcal{O}_X=0$ for $i>0$? (I do not need the complete ...
4
votes
1answer
33 views

What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles?

I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold ...
2
votes
1answer
41 views

Separated Schemes and Intersection

Let $X$ be a separated scheme. I am trying to show that if $U$ and $V$ are affine open sets then $U\cap V$ is also. I can see that $U\cap V$ is homeomorphic to $d(X)\cap (U\times V)$. Where $d$ is the ...
1
vote
0answers
47 views

Why is the codimension of an algebraic set defined by $r$ equations at most $r$?

Suppose I have $r$ polynomials $g_1, ..., g_r$ in $\mathbb{Z}[x_1, ..., x_n]$. And let $H = \{ \mathbf{x} \in \mathbb{C}^n : g_i(\mathbf{x}) = 0 (1 \leq i \leq r) \}$. I was wondering why it then ...
3
votes
1answer
34 views

Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
1
vote
1answer
50 views

Addition of points on elliptic curves over a finite field

I have found the following formulas for the coordinates of $P+Q$ given that $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ are points on a general curve $y^2 = x^3 + ax + b$ over $\mathbb{R}$: $$P + Q ...
2
votes
2answers
30 views

Basic question related to the definition of affine $k$- variety

The definition of affine $k$- variety $X$, I have is that $X$ is an affine scheme that is reduced and of finite type over $k$ ($k$ is a field here). The definition of finite type I have is that $X$ ...
2
votes
1answer
56 views

Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves

I want to read the proof of the following theorem: This is from p.35. But it is not complete there. There is written that: Can someone tell me where I can find the rest of the proof? Any other ...
0
votes
0answers
13 views

Birational map between a conic and an affine line (related to the classical formula of Pythagorean triples)

Could someone please explain me the following? It says in the notes I am reading that $$ Spec \ (\mathbb{Q}[x,y] / (x^2 + y^2-1) ) \rightarrow Spec \ \mathbb{Q}[m] $$ given by $$ f : (x,y) ...
2
votes
2answers
41 views

Show that the ideal $I=\left\langle x_1^2+1,x_2,…,x_n\right\rangle$ is maximal in $\mathbb{R}[x_1,…,x_n]$.

This is an exercise in "Ideals, varieties, and algorithms" by Cox et al. It first asks to show that $I=\left\langle x^2+1\right\rangle$ is maximal in $\mathbb{R}[x]$. I can show it because it is a ...
4
votes
2answers
112 views

Modular curves over finite fields

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
5
votes
1answer
30 views

Why $R^q(\Gamma \circ \eta_{*}) (\Bbb G_{m, \eta}) = H^q(\eta_{ét}, \Bbb G_{m, \eta})$?

Let $X$ be a smooth, projective and connected curve over an algebraically closed field, and let $\eta \rightarrow X$ be its generic point (we also call the inclusion as $\eta$). I want to understand ...
3
votes
2answers
99 views

How is the multiplicative group an algebraic variety?

According to various places, we define an algebraic group as a group that is also an algebraic variety (along with some compatibility conditions). Many places also list some examples, one of which is ...
0
votes
1answer
42 views

Calculating intersection number of $(x^2+y^2)^3-4x^2y^2=0 $ and $x=0$ at $(0,0)$

I am trying to find the intersection number of $(x^2+y^2)^3-4x^2y^2=0 $ and $x=0$ at $(0,0)$. The intersection number of $F$ and $G$ is defined to be $dim_k(O_p/(F,G))$(Here $O_p$ is the local ring ...
4
votes
1answer
64 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
1
vote
0answers
49 views

Generalizing points on the x, y, and z planes

I am having a little trouble developing the intuition to understand where the points $P = (x,y,z)$ in $\mathbb{R}^3$ with planes only. For instance, the equation $ xyz = 0 $ represents just the ...
1
vote
1answer
31 views

Given a point $P$ and a hyperplane $H$ in $\mathbb{P}^n$ such that $P \in H$, there is $T$ linear such that $T(P)=(0:\cdots:0:1)$ and $H:X_0=0$

Show that given a point $P$ and a hyperplane $H \subseteq \mathbb{P}^n$ such that $P \in H$, there is a linear transformation $T$ such that $T(P)=(0:\cdots:0:1)$ and $H$ is given by the equation ...
2
votes
1answer
45 views

Calculating the coordinate ring and irreducible components

Consider the graded ring $S=(R/I)\oplus (I/I^2)\oplus (I^2/I^3)\oplus\cdots$ Take $R=k[X,Y],I=(X^2Y,XY^2)$. Then $S=k[X,Y]/(X^2Y,XY^2)\oplus(X^2Y,XY^2)/(X^2Y,XY^2)^2\oplus\cdots$. I am not sure ...
0
votes
0answers
53 views

Product of Schemes and Open Subsets

Let $X$ be a scheme and $U$ an open subset, view $U$ as a scheme also. Let $X\times X$ be the product in the category of schemes. Show that there exists an open subset $V$ of this product, such that ...
1
vote
0answers
36 views

Difference between quadric and conic

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...
0
votes
0answers
42 views

An integrally closed subdomain of a polynomial ring

Let $\mathbb{C} \subset R \subset \mathbb{C}[x,y]$ be a noetherian integral domain. Further assume that: (1) $\mathbb{C}[x,y]$ is separable over $R$. (2) $\mathbb{C}[x,y]$ is algebraic over $R$ ...
2
votes
0answers
30 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
3
votes
1answer
41 views

Zero-section as homomorphism of rings

Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated ...
0
votes
0answers
52 views

Computation of Riemann-Roch space L(kQ) to a specific Divisor D

I am trying to build a Reed-Solomon Code through a Goppa-Code Construction. I start with the projective line $\mathcal{X}$ $aX+bY+cZ=0$. The genus $g$ of this line is $0$. Futhermore, let ...
1
vote
1answer
33 views

Morphisms induced by effective divisors on $\mathbf P^1$

This question is about the proof of Theorem V.2.17 in Hartshorne's Algebraic Geometry. Here everything is defined over some algebraically closed field $k$. Define $\mathcal O = \mathcal O_{\mathbf ...
1
vote
1answer
44 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
4
votes
1answer
58 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
4
votes
2answers
72 views

Dimension of the affine variety associated to $\langle zw-y^2, xy-z^3 \rangle $

Find the dimension of the affine variety $V(I)$, where $I=\left\langle zw-y^2,xy-z^3\right\rangle \subseteq k[x,y,z,w]$, with $k$ algebraicaly closed field. I tried to solve the system $zw-y^2=0$, ...
3
votes
1answer
35 views

base change of an equivalence relation of fppf sheaves

Let $S$ be a scheme, $R,U$ be $S$-schemes and $s,t : R \to U \times_S U$ be an equivalence relation i.e. it's a monomorphisme such that for every $S$-scheme $T$, $R(T) \to U(T) \times U(T)$ is and ...
5
votes
1answer
58 views

Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the ...
0
votes
0answers
25 views

Applications of projective normality

Why study projective normality of a variety ? What are the applications ? How does it relate to non-singularilty, rationality etc of the variety ?
1
vote
1answer
33 views

With regards to Theorem 3.2 in Hartshorne: Are regular functions on a variety simply polynomials?

I am reading Hartshorne's Algebraic Geometry for the first time and I am having some trouble understanding Proposition 3.2. The proposition implies that $\begin{array}{ccccc} \mathcal{O}(Y) ...
1
vote
1answer
32 views

Weierstrass normal form of an elliptic curve

without knowing any deeper theory, I am required to find the Weierstrass normal form of an elliptic curve, i.e. a representation of type $y^2z-x^3-axz-bz^3$ where $x,y,z $ are variables and $a,b$ are ...
0
votes
1answer
41 views

Standard Cremona Involution

Let $\varphi$ be the standard Cremona involution on $\mathbb{P}^r$, which is defined as $[x_0,\dots,x_r]\mapsto [\frac{x_0\dots x_r}{x_0},\dots, \frac{x_0\dots x_r}{x_r} ]$. I came across to the ...
0
votes
1answer
32 views

Closure of Set in Zariski Topology [on hold]

If $U$ is a set in affine space, is the closure of $U$ simply $V(I(U))$? If so, how might one prove this?
4
votes
0answers
37 views

Birational map between reduced schemes.

I am reading Ravi Vakil's notes "Foundation of Algebraic Geometry" and in Proposition 6.5.5., it states the following: Suppose $X$ and $Y$ are reduced schemes. Then $X$ and $Y$ are birational if and ...
1
vote
0answers
27 views

filtration of vector bundle

Suppose I have a vector bundle on a variety. When is it true that my vector bundle admits a filtration with subquotients all line bundles? (I am most interested in the case of an integral projective ...
0
votes
0answers
26 views

Discriminant ideal and short exact sequence of finite group schemes

Let $0 \to G' \to G \to G'' \to 0$ be a short exact sequence of finite flat commutative group schemes over a Dedekind domain $\mathcal O$ with field of fractions of characteristic zero. Let ...
3
votes
1answer
147 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
1
vote
1answer
68 views

General Structure Sheaf Question

In reading the Red Book of Varieties and Schemes, I am confused on this idea boxed in the image: . Why does $F(x)=0$ for all $x \in U$ imply $F=0$? Can't we only say this if $F$ vanishes on all of ...
1
vote
1answer
39 views

Singular plane cubic curve birational to $\mathbb{P}^1$

Is it true that every singular plane cubic curve over an algebraically closed field is birationally equivalent to $\mathbb{P}^1$? I know that such a curve has to have only one singular point and that ...
10
votes
2answers
382 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 ...
2
votes
2answers
39 views

Isomorphism of Varieties

Let $V=V(x^2+y^2-1) \subset \mathbb{R}^2$ be an affine variety. Show that $V$ is rational, but isn't isomorphic to $\mathbb{R}^1$. I could show that $V$ is rational, by parametrization ...
8
votes
1answer
103 views

Is there a “Coalgebra - Cogeometry” duality? Good opposite of a category of coalgebras?

So the category of affine schemes is dual to the category of commutative rings, Stone spaces are dual to Boolean algebras, localizable measurable spaces are dual to commutative Von Neumann algebras, ...
1
vote
1answer
42 views

Finite Variety in $\mathbb{C}^5$

Let $V=V(x_3-x_1^2,x_4-x_1x_3,x_2x_3-x_1x_5,x_4^2-x_3x_5)\subseteq \mathbb{C}^5$ be an affine variety. Is V a finite set of points? I tried using Groebner bases, but I can not get anywhere. Could ...
1
vote
0answers
34 views

Connection between local freeness and the rank of matrices

I am reading ch.16 of Eisenbud's Commutative Algebra, more precisely it's the very first paragraph of 16.7, where he wants to prove: Suppose that $\mathcal{J}: R^t \longrightarrow R^r$ is a map of ...
1
vote
0answers
20 views

Algebraic cusp-planar curves

Quote from Wikipedia:"The plane curve cusps are all diffeomorphic to one of the following forms: $x^2 − y^{2k+1} = 0$, where $k ≥ 1$ is an integer." 1)Can one provide a reference for this? ...
1
vote
0answers
26 views

The fibers of a ruled surface form an algebraic family of divisors

I'm reading Chapter V.2 of Hartshorne, which includes the following claim, where $\pi: X \rightarrow C$ is a ruled surface: Note that any two fibres of $\pi$ are algebraically equivalent divisors ...
4
votes
1answer
781 views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
9
votes
2answers
81 views

Nonsingular curve $C$ of degree 4, exists rational function $f: C \to \mathbb{CP}^1$ of degree 2?

Suppose $C \subset \mathbb{CP}^2$ is a nonsingular curve of degree $4$. Does there exist a rational function $f: C \to \mathbb{CP}^1$ of degree $2$?