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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2
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15 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
1
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0answers
29 views

Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
3
votes
0answers
46 views

Evaluate a rational function at infinity

In the context of the Tate pairing, I would like to know that it means to `evaluate' an $\mathbb{F}_{q^k}$-rational function at $\infty$. For instance, the reduced Tate pairing is $e_n:G_1\times ...
1
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0answers
44 views

Question related to the definition of affine schemes

The definition for affine schemes I have learned was that it is a locally ringed space $(X, O_X)$ that is isomorphic to $(Spec \ A, O_{ Spec \ A })$ for some ring $A$ (commutative and includes $1$). ...
1
vote
1answer
22 views

How to calculate 2-d plane from 3 4-d points?

I want to compute 3-d cross-sections of a pentatope (4-dimensional tetrahedron). The 3-d cross-sections will be calculated as: x+y+z+w=c C is a constant that I will vary to get different ...
1
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2answers
69 views

Is there a classification of regular maps $\mathbb{P}^1(k)\to\mathbb{A}^1(k)$?

If $\mathbb{P}^1(k)$ and $\mathbb{A}^1(k)$ are the projective line and affine line, respectively, over an algebraically closed field $k$, is there any known classification of the regular maps ...
4
votes
2answers
65 views

Singular Points on Algebraic Curves

What does it mean for a point on an algebraic curve (over any field) to be singular? Over $\mathbb{R}$ or $\mathbb{C}$ is means that all derivatives vanish, but how does this definition generalise to ...
3
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0answers
39 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
2
votes
1answer
50 views

Base change for Quot-scheme

I am reading the book of Huybrechts and Lehn "The Geometry of Moduli Spaces of Sheaves" with an aim to become a little bit familiar with this topic. Now I am trying to understand what is ...
4
votes
1answer
84 views

Cohomology of affine plane with double origin

How to calculate cohomology $H^1(X,O_X)$,$H^2(X,O_X)$ $H^1(X,O_X^*)$ of affine plane with double origin $X=\mathbb{A}^2\cup_{\mathbb{A}^2-\{0\}}\mathbb{A}^2$? To use Cech cohomology, I cannnot find a ...
2
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1answer
46 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
2
votes
1answer
50 views

direct and inverse images of sheaves and some canonical morphisms

Consider a continuous map $f\colon X\to Y$ between topological spaces. Let $\mathcal F$ be a sheaf on $X$ and $\mathcal G$ a sheaf on $Y$ (let's say of abelian groups). There exists canonical ...
0
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0answers
34 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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0answers
32 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
0
votes
1answer
27 views

Show that $\mathbf{I}(\mathbf{V}(x^n, y^m)) = \langle x, y \rangle$.

Show that $\mathbf{I}(\mathbf{V}(x^n, y^m)) = \langle x, y \rangle$. Where $\mathbf{I}$ is the ideal, and $\mathbf{V}$ is the affine variety. I'm not sure how to even begin on this one. I know ...
0
votes
0answers
63 views

immersions and finite morphisms

I have the following question: Let $X \subset \mathbb A^n$ be an affine variety. Prove that the immersion $i\colon X \hookrightarrow \mathbb A^n$ is a finite morphism. I know that the ...
2
votes
2answers
49 views

Restriction maps in an integral scheme are injective

Suppose $X$ is an integral scheme. I would like to show that the restriction maps $res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could ...
1
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1answer
63 views

Why do intersection of two quadratic forms implies elliptic curve?

Let $k$ be a field and $S=k[T_0,T_1,T_2,T_3]$ and $f,g\in S_2$ two relatively prime quadratic forms. How can I show that the intersection $X\subset \mathbb P_k^3$ of second degree surfaces $V_+(f)$ ...
3
votes
2answers
104 views

Why Zariski topology is not Hausdorff

I am reading the book about Algebraic geometry. I am confused about the following two things the book mentioned: Zariski topology is 1. different from the topology studied in real and complex ...
0
votes
1answer
30 views

Sparse & Dense Polynomials

I've been reading up on Bernstein's theorem for an algebraic geometry assignment and I've come across the terms "dense" and "sparse" in relation to the polynomials. However, I have been unable to find ...
2
votes
1answer
29 views

Showing a set is an affine variety

I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following: Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set ...
1
vote
1answer
35 views

suggestion for a book on algebraic curves

I don't know if i'm asking in the wrong place, but I'm studying the book Algebraic curves of Fulton, and having some problems understanding the final chapters (6,7 and 8). I've found a good help in ...
2
votes
1answer
39 views

Basic question related to sheaf of a scheme

Suppose I have a scheme $X$. And some non-empty open set $U \subseteq X$. Does it then follow that $O_X(U)$ is not the trivial $0$-ring by any chance?
2
votes
1answer
63 views

Regular Local Ring

Let $Y$ be an affine variety in $\mathbb{A}^n_k$ and $\mathfrak{i}$ its corresponding ideal. We use the notation $A(Y) = k[x_1,...,x_n]/\mathfrak{i}$ for the coordinate ring of $Y$. Pick a point $p\in ...
1
vote
1answer
54 views

Why schemes are $(X,\mathcal O_X)$ rather than $(\mathcal O_X,X)$ or $\{X,\mathcal O_X\}$

Is there a reason why schemes are ordered pairs $(X,\mathcal O_X)$ rather than for example $(\mathcal O_X,X)$ or $\{X,\mathcal O_X\}$?
8
votes
1answer
116 views

Does there exist a regular map $\mathbb{A}^1\to\mathbb{P}^1$ which is surjective?

Suppose $\mathbb{A}^1$ and $\mathbb{P}^1$ are affine space and projective space. I'm not sure if it matters, but I don't mind if we assume that we're working over algebraically closed fields. I'm ...
3
votes
0answers
22 views

Closed subset of a affine linear group [duplicate]

Let $G\subseteq GL_n(\mathbb{C})$ a Zariski-closed linear subgroup and $X\subseteq G$ closed with $X*X\subseteq X$ and $e \in X$. Then $X$ is a subroup. I am not sure how to start here. I know that ...
3
votes
1answer
49 views

Calculating canonical divisor in product of projective spaces.

Let $X$ be an intersection of two divisors of bidegree $(a,b)$ and $(c,d)$ in $\mathbb{P^2}\times \mathbb{P^2}$. Then how can I find the canonical divisor $K_X$? I'm asking because I have no ...
2
votes
0answers
24 views

Decomposition of abelian varieties up to isogeny

Let $A_1,A_2,B_1,B_2$ be simple abelian varieties over a number field $k$. Suppose that $A_1\times A_2$ is $k$-isogenous to $B_1\times B_2$. Can we deduce that (up to reordering the factors) $A_1$ is ...
4
votes
2answers
44 views

pullback of twisting sheaf

Let $[k]: \mathbf{P}^n \to \mathbf{P}^n, [x_0:\ldots:x_n] \mapsto [x_0^k:\ldots:x_n^k]$ be a morphism. (Why) do we have $[k]^*\mathcal{O}_{\mathbf{P}^n}(1) \cong \mathcal{O}_{\mathbf{P}^n}(k)$?
0
votes
0answers
23 views

In P^n(projection of C^n+1) is a variety isomorphic to P^1 irreducible?

In P^n(projection of C^n+1) is a variety isomorphic to P^1 irreducible? I think not because is the union of a line and a point at infinity
1
vote
1answer
40 views

Quillen's K-theory and ring homomorphisms

I am a beginner in algebraic K-theory and I want to make sure that I understand the following correctly: Let $f:A \to A'$ be an isomorphism of commutative rings. Denote by $\mathcal{P}(A)$ (resp. ...
2
votes
0answers
45 views

On a method to compute dimension of moduli space of Riemann surfaces

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
0
votes
1answer
39 views

local intersection multiplicity

I am reading kenji uneoگس book on algebraic geometry 1. I don't understand how to compute the local intersection multiplicity. I would appreciate if you can show me how to compute it for the next two ...
1
vote
1answer
41 views

Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
1
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0answers
42 views

Cartier divisors of schemes

In his notes, Ravi Vakil only defines the notion of an effective Cartier divisor. Furthermore, the Wikipedia page only defines the notion of an effective Cartier divisor for a general scheme. ...
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0answers
21 views

What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction. At the heart of this is the complex $$ V ...
2
votes
0answers
28 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
9
votes
2answers
170 views

Artin Chapter 11, Exercise 9.12, polynomials without common zeroes [closed]

How do I show that the three polynomials $f_1 = t^2 + x^2 - 2$, $f_2 = tx - 1$, $f_3 = t^3 + 5tx^3 + 1$ generate the unit ideal in $\mathbb{C}[t, x]$? Artin mentions two approaches: by showing that ...
7
votes
1answer
79 views

Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
1
vote
1answer
58 views

Simple question about the traslation from french to english

Goodmorning. I'm reading an article by Arnaud Beauville talking about the surfaces of general type. I've found this term and i'm not sure about the translation: What does it mean a " pinceau de ...
0
votes
0answers
49 views

Book/lecture notes on algebraic curves

Although there surely is plenty of references on MSE about algebraic curves, my need are very specific and so I will open this topic anyway. I follow this year a course on (hyper)elliptic curves ...
0
votes
1answer
27 views

twists of unipotent algebraic groups

Let $U$ be a unipotent linear algebraic group over some field $k$ with char$k$=0. Let $U'$ be a linear algebraic group over $k$ such that $U'_{\bar{k}} = U_{\bar{k}}$ (ie $U'$ is a $\bar{k}/k$-twist ...
3
votes
2answers
68 views

Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial.

I am working on a problem from Ideas, Varieties, and Algorithms: Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero ...
2
votes
1answer
44 views

Blow Up of a Variety

Let $Y$ be an affine variety of $\mathbb{A}^n$ and it contains $0$. Think of $\mathbb{A}^n \times \mathbb{P}^{n-1}$ as a quasi-projective variety. Define a closed subset of $\mathbb{A}^n \times ...
3
votes
1answer
72 views

Castelnuovo-Mumford regularity of Cohen-Macaulay modules

Let $S=K[X_1,\ldots,X_n]$ and $M$ be a Cohen-Macaulay $S$-module. This equality holds $$ \operatorname{reg}(M)=\dim(M)+\max\{i\in\mathbb{Z}\colon P_{M}(i)\neq H(M,i)\}. $$ It's been proved in ...
3
votes
1answer
38 views

isomorphism of pointed sets

What is an isomorphism in the category of pointed sets? Is it just an exact sequence $$ 1 \to A \to B \to 1 ?$$ (Note: even though the kernel of the middle map is zero, $A$ might not inject into $B$.) ...
1
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1answer
56 views

Book recommendations for topics leading upto Algebraic geometry

I'm interesting in studying algebraic geometry (specifically either from Shafarevich or Hartshorne). Assuming a high school and basic college math education, what should be the topics and the order ...
2
votes
1answer
50 views

Abstract Varieties

Hartshorne does not seem to bring this concept up so far in his AG book but I am guessing that one may define an "abstract variety", in a similar way as one defines an abstract manifold from DG. ...
2
votes
1answer
36 views

What is the dimension of $A-B$, where $B$ is a subspace of $A$?

My question is really simple, what is the dimension of $A-B$, where $B$ is a subspace of $A$? this space is well-defined? I found this space in this paper on page 440: Following my calculations in ...