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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Proof that Distinguished Open Set is an Affine Variety

I'm trying to understand the proof in Mumford's Red Book that a distinguished open set of an affine variety is itself an affine variety. I've attached an image of the proof, and cannot seem understand ...
2
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0answers
21 views

Locally free sheaves on $\mathbb{A}^n$-bundles

Let $X$ be a variety, and let $p: E \rightarrow X$ be a $\mathbb{A}^n$-bundle. By this, I mean there is an open cover $U$ of $X$ such that if we base change to the open cover, $E \times_X U \simeq U ...
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1answer
35 views

Ring of regular functions on a point

Let $X \subset \mathbb{A}^n$ be an affine variety. Then the ring $\mathcal{O}_X$ of regular functions on $X$ is $A(X) := k[y_1,\dots,y_n] / I_X$, where $I_X$ is the vanishing ideal of $X$ (and $k$ is ...
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11 views

Question on cone of projective algebraic set $V$

Suppose I have a projective algebraic set $V \subseteq \mathbb{P}^n$ and its cone $C(V) \subseteq \mathbb{A}^n$. I was wondering about the following statement: $V$ is irreducible if and only if $C(V)$ ...
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9 views

Reference on Malcev completion

I need a reference for learning Malcev completion, its associated group scheme and Lie algebra. Thanks!
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36 views

Is there a natural map $\Omega^1_X\to N^{\vee}_{Y/X}$?

Let $X$ be a smooth complex manifold and $Y\subset X$ be a complex submanifold. Is there some natural map from $\Omega^1_X$ to $N^{\vee}_{Y/X}$?
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1answer
16 views

Find order of elliptic curve

Given a prime $p$ such that $3$ does not divide $p-1$, what is the order of the elliptic curve over $\mathbb{F}_p$ given by $E(\mathbb{F}_p)=\{ (x,y) \in \mathbb{F}_p^2 | y^2=x^3+7 \}$ I thought if ...
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30 views

Is the group-theoretic Grothendieck-Springer resolution Calabi-Yau?

Any cotangent bundle is Calabi-Yau (by which I mean the canonical bundle is trivial), so the Springer resolution $T^*(G/B)$ is Calabi-Yau. I think that the Grothendieck-Springer resolution ...
4
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1answer
21 views

Existence of a Cech cover for computing Picard group

Let $X$ be a variety -- one can compute $\text{Pic}(X) = H^1(X, \mathcal{O}^*_X)$ by choosing a Cech cover which is acyclic with respect to $H^\bullet(-, \mathcal{O}^*)$. Can one always do this? It ...
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2answers
362 views

What *is* affine space?

In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$: $\mathbb{A}_k^n$ is $k^n$ 'without ...
2
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0answers
72 views

What does Hartshorne mean here in Proposition 2.3?

I was re-reading Hartshorne, proposition 2.3 on page 73. He says: Now for any open set $V\subset \operatorname{Spec}A$ we obtain a homomorphism of rings ...
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2answers
105 views

Two questions on the Grothendieck ring of varieties

1) In the definition of the Grothendieck ring of varieties over a field $k$, which definition of the various notions of "variety" is chosen? Finite type and separated, or maybe more? 2) If ...
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1answer
43 views

Projections of the twisted cubic curve from points

I. Consider the twisted cubic curve $C$ in $\mathbb{P}^3$, given as the image of the veronese map $v_3: \mathbb{P}^1 \rightarrow \mathbb{P}^3$. Let $p \in \mathbb{P}^3 $ and consider the projection ...
3
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0answers
66 views

A basic question about closed set in Zariski topology

Suppose I have homogeneous polynomials $f_1, .., f_r \in \mathbb{C}[x_1, ..., x_n]$, and let $I = (f_1, ..., f_r)$. Let $V:=V(I) \subseteq \mathbb{C}^n$ be the points where $f_i$'s vanish. Suppose $V$ ...
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1answer
40 views

Finding the Krull dimension of a quotient ring $\mathbb{C}[x,y,z]/I$

I am interested in finding the Krull dimension of the quotient ring $A$ defined as follows: $$ A = \mathbb{C}[x,y,z] / (f_1, f_2, f_3), $$ where $$ f_1 = \frac12 y^3 z - (z-1) - xy $$ $$ f_2 = y^2 z^2 ...
2
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0answers
32 views

On a sufficient condition for a closed morphism of schemes to be affine

Let $f \colon X \to Y$ be a closed morphism of schemes (i.e., the image of any closed subset of $X$ under $f$ is closed in $Y$). Let $y \in Y$. Consider the following assertions: (i) There is an ...
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1answer
31 views

Question about basic properties of degrees of algebraic sets

I am learning about degrees of algebraic sets at the moment, and in an article I am reading I came across the following: Let $V_i \subseteq \mathbb{C}^n$ be a hypersurface of degree at most $D$ for ...
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1answer
24 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
3
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0answers
63 views

Equation of 27 lines on a cubic surface [on hold]

For a smooth cubic surface $S$ in $\mathbf{P}^3$, there're always $27$ lines on it, with the same configuration. We know the automorphism group of the lines is not solvable. How do we show the ...
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0answers
88 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
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2answers
73 views

What is the convention for the codimension of an empty set?

I am learning about dimension and codimension of algebraic sets at the moment. I know that if $V \subseteq \mathbb{C}^n$ is an algebraic set defined by polynomials $f_1, ..., f_r \in \mathbb{C}[x_1, ...
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0answers
43 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
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2answers
83 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
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0answers
23 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
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1answer
32 views

Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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54 views

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curves second edition (J. H. Silverman) [closed]

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curve second edition (J. H. Silverman)
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0answers
51 views

How to obtain the genus of the Riemann surface corresponding to an algebraic curve

I am trying to read about the genus of an algebraic curve. I have been told that there is a connection between topological genus and genus defined for an algebraic curve. Since an algebraic curve ...
1
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1answer
85 views

Basic question related to dimension of intersection of two varities

Let $V$ and $W$ be irreducible varieties in $\mathbb{C}^n$. I have learned that intersection $V \cap W$ satisfies the following: $$ codim \ V + codim \ W \geq codim \ V \cap W. $$ I was wondering if ...
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85 views
+200

Why is the morphism induced by this linear system birational?

I have seen (what seems to be) the following statement used in a few places, but I am not sure why it is true. Any explanation as to why it is (or is not) true would be appreciated. Let $P$ be a ...
0
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1answer
96 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
2
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0answers
26 views

Basic question about the properties of dimension of an algebraic set

I am learning about the dimension of an algebraic set and I have a couple of questions I am hoping to resolve to have a better understanding. Let $V$ be an algebraic set in $\mathbb{C}^n$, defined ...
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0answers
37 views

How to show that $\mathcal{K}/\mathcal{O}$ is isomorphic to$\sum_{P\in X} i_P(I_P)$

This a Hartshorne exercise (Ex II 1.21d) Let $X=\mathbb{P}^1$. Let $\mathcal{K}$ be the constant sheaf of the quotient field of X. Then we need to show that $\mathcal{K}/\mathcal{O}$ is isomorphic to ...
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36 views

Show that exist point $w \in \mathbb{C}$ with $w^2 \in K$ and $z \in K(w)$

Let K is subfield of $\mathbb{C}$ with $K=\overline{K}$. $F(k)$ is a set with all circles in complex plane with midpoint in K and radi^us equal distance between two points from K. Let $z\in ...
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0answers
15 views

Group action on closed subschemes

Let $G$ be a connected, linear, semi-simple algebraic group and $P \subset G$ the maximal parabolic subgroup. We know that $Z=G/P$ is a projective variety. Then, 1) Does $Z$ contain a line? 2) In ...
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0answers
40 views

Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that Let $X$ be a normal variety, ...
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1answer
102 views
+50

Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
5
votes
1answer
56 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
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0answers
12 views

Radical of reductive but not connected linear algebraic groups

Let $G$ be a linear algebraic group over a field $k$ of characteristic zero. A definition of $G$ being reductive is that the radical of $G^0$ (the connected component of the identity of $G$) over ...
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1answer
14 views

connection between absolute irreducibility and smooth+geometrically connected

Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon ...
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2answers
111 views
+50

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
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37 views

Definition of regular functions on a projective variety

I'm trying to read Algebraic Geometry : a First Course by Joe Harris. In Lecture 2, p. 20, he defines a regular function on an open set $U$ of quasi-projective variety $X$ as a function such that if ...
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43 views

Isolated points of fibers of regular morphism

Let $X,Y$ be affine varieties and $\varphi:X\to Y$ be regular morphism. I want to prove that isolated points of fibers of $\varphi$ form open subset in $X$. Can you give me advice how to do it?
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1answer
68 views

How to show that a map is finite

Let $V = Z\left(X^3 - Y^2\right)\in \mathbb{k}^2$. How to show that $f \colon t \in \mathbb{C} \mapsto \left(t^2, t^3\right) \in V$ is a finite map? Thanks in advance!
2
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44 views

Automorphism of $\mathbb{A}^2$ which maps the finite set of points to the finite set of points

Let $\mathrm{k}$ be infinite field. $P_1,\dots,P_n, Q_1,\dots,Q_n \in \mathbb{A}^2$ and $P_i \neq P_j, Q_i \neq Q_j$. I want to find automorphism(in a.g. sense) which maps $P_i$ to $Q_i$. I have tried ...
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1answer
189 views

A $X \subseteq \mathbb{A}^n$ such that $I(X) \neq I(V(I(X)))$?

Let $\mathbb{A}^n$ be the affine $n$-space over a field $K$. Denote by $V(S)$ the zero locus of a $S \subseteq K[x_1, \dots, x_n]$ and let $I(X)$ be the ideal of a $X \subseteq \mathbb{A}^n$. Is there ...
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0answers
54 views

Prerequisites “Homology Theory of Algebraic Varieties” by Wallace

I bought this book because the title was very interesting, the description as well and the price very cheap. You can read it here in PDF. Unfortunately, I realized after reading the first lines I was ...
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1answer
31 views

Representable morphism for algebraic spaces

I'm trying to understand the definition of algebraic spaces, but there is a notion of representable morphism that is a little confusing to me. Let $S$ be a scheme and let $Sch/S$ denote the category ...
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2answers
48 views

On the definition of degree of a hypersurface

Let $f \in \mathbb{C}[x_1, ..., x_n]$ be a homogeneous polynomial of degree $d$. I was trying to understand the definition of degree of hypersurfaces. It says on Wikipedia ...
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
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1answer
43 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 ...