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)

1
vote
1answer
22 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 ...
1
vote
1answer
26 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 ...
3
votes
0answers
62 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 ...
11
votes
2answers
99 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 ...
2
votes
0answers
17 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 ...
0
votes
1answer
83 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 ...
0
votes
1answer
38 views

Fulton, algebraic curves exercise 4.11

I'm doing the exercises of the book Fulton Algebraic curves and I'm stucked in the following problem: A subset $V\subset\mathbb{P}^n(k)$ is a linear subvariety of $\mathbb{P}^n(k)$ if ...
15
votes
0answers
197 views
+100

Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
11
votes
1answer
497 views

Gaining insight into the Inverse Image Sheaf

Let $f: X \rightarrow Y$ be a continuous map of topological spaces and let $G$ be a sheaf of sets on $Y$. I am trying to understand the definition of the inverse image sheaf $f^{-1}G$ on $X$. This is ...
5
votes
1answer
75 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
4
votes
0answers
63 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.
2
votes
1answer
104 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
1
vote
3answers
64 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, ...
0
votes
1answer
19 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
votes
0answers
50 views

Equation of 27 lines on a cubic surface

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 ...
3
votes
1answer
58 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
1
vote
0answers
25 views

Proof of 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 ...
0
votes
2answers
54 views

Endomorphisms in an exact sequence of vector bundles

Let X be a smooth projective variety over $\mathbb{C}$. And suppose we have an exact sequence of vector bundles over $X$. $\qquad\qquad\qquad\qquad\qquad 0\longrightarrow A\longrightarrow ...
7
votes
0answers
140 views
+50

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
2
votes
2answers
75 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 ...
1
vote
1answer
82 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 ...
0
votes
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 ...
1
vote
1answer
31 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 ...
1
vote
1answer
45 views

Examples of base points of linear systems

I'm reading Fulton's algebraic curves book and we have the following definitions: A divisor $D=n_1P_1+\ldots,n_kP_k$ ($n_i$'s are integers and $P_i$'s are points) over a curve. A linear system as ...
1
vote
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} ...
2
votes
2answers
42 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
5
votes
2answers
174 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
-4
votes
0answers
44 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) [on hold]

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)
0
votes
0answers
45 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 ...
2
votes
0answers
25 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 ...
0
votes
0answers
36 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 ...
2
votes
1answer
84 views

Finding a stronger version of Cayley-Bacharach Theorem that applies in the case that the intersection multiplicities are not equal to $1$

Cayley–Bacharach theorem: Assume that two cubics $C_1$ and $C_2$ in the projective plane $\mathbb{P}^2$ meet in nine (different) points. Then every cubic that passes through any eight of the points ...
0
votes
0answers
35 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 ...
0
votes
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 ...
1
vote
0answers
38 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, ...
5
votes
1answer
55 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 $\pi_1(X-Z)=\pi_1(X)$ ...
2
votes
0answers
43 views

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 ...
0
votes
0answers
10 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 ...
2
votes
0answers
34 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 ...
1
vote
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!
3
votes
0answers
41 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?
13
votes
2answers
1k views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
-3
votes
1answer
184 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 ...
2
votes
0answers
43 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
2answers
46 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 ...
6
votes
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 ...