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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43 views

The same algebraic variety defined by different sets of polynomials

Let $\emptyset\neq X\subset\mathbb{P}^{n}$ be an algebraic variety such that $$ X=V(F_{1},\ldots,F_{m}) $$ for certain linearly independent homogeneous polynomials $F_{1},\ldots,F_{m}\in ...
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1answer
44 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
3
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52 views

About birational map between curves of the form $y^{2}=g(x)$

I'm trying to solve the following exercise but I'm stuck after trying for a long time. Suppose that $g(x)=ax^{4}+bx^{3}+cx^{2}+dx+e\in{k[x]}$ and similarly ...
1
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1answer
33 views

Ideal ring of polynomials in two variables with real coefficients

Let $ I= \langle x ^ 4 + y ^ 4 + 2x ^ 2y ^ 2-x ^ 2-y ^ 2 \rangle \subset\mathbb R[X,Y]$. I want to determine whether $ I $ is prime or radical. I know that $I$ is not prime. First, $ \langle x ^ 4 + ...
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0answers
32 views

Extension of Leray spectral sequence, Vakil's 23.4 H

If you have a morphism \begin{equation} (X,\mathscr{O}_X) \xrightarrow {\pi} (Y, \mathscr{O}_Y) \end{equation} for every $\mathscr{O}_X$-module $\mathscr{F}$, there is a spectral sequence with $E_2$ ...
2
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2answers
63 views

Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
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0answers
9 views

How to determine the angle from a point and the plane tangent points in a sphere

I have an UAV modeled in three dimensions with let's say position coordinates $p_{uav} = (x_1,y_1,z_1)$ that is moving in a direction $d = (d_x,d_y,d_z)$ and a moving obstacle modeled as a sphere with ...
3
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0answers
66 views

Proof of Chapter 2 Proposition 2.6a in Silverman Arithmetic of Elliptic Curves

The following is Proposition 2.6(a) in Silverman's AEC: "Let $\phi: C_1 \rightarrow C_2$ be a nonconstant rational map of smooth (projective) curves over an algebraically closed field $K$. Then ...
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1answer
15 views

Number of zero-solutions for two bivariate polynomials $p$ and $q$

If I consider two bivariate polynomials $p,q \in \mathbb{C}\left[ x,y \right]$ where $p$ has total degree $m$ and $q$ has total degree $n$. To keep things simple I'm not interested in special cases ...
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1answer
37 views

Relation between Homeomorphisms and Isomorphisms for varities.

I am right now learning Algebraic Geometry and at the first moment is very demanding. One of my biggest doubts is: why algebraic geometers despise so much homeomorphisms , all books that I have been ...
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0answers
22 views

Self-intersection of an axis

Let $X$ be a projective, smooth curve over an algebraically closed field, $Y = X\times X$ and $\mathcal{l}= pt\times X$ where $pt$ is a closed point of $X$. Can one say that the intersection number ...
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0answers
22 views

Volume question regarding segmenting a truncated cylinder.

Picture of a truncated wedge segment 2If you segmented a truncated cylinder, ensuring all segments had the same volume, where would the intersection be? I'm understand there'll probably not be a ...
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0answers
23 views

Brill-Noether theory- reference request

I need some reference, some books or something suitable for a begginer. I found some .pdf's on google, that have interesting introduction, but couldn't find any book.
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1answer
56 views

Chern class of ideal sheaf

Let $X$ be a smooth projective surface. Let $Z$ be a dimensional $0$ subscheme of length $l$. Suppose $I_Z$ is the ideal sheaf of $Z$. Then it claimed that $c_1(I_Z) = 0$ and $c_2(I_Z) = l$. (1)Why ...
3
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1answer
60 views

Hilbert function and homogenous polynomials.

Let $\{[1:0:0],[0:1:0],[0:0:1],[1:1:1] \} = \{p_1,p_2,p_3,p_4\}$ be four points in the projective space $\mathbb{P}^2$. For every $p_i$, show there is a homogenous polynomial $f_i$ such that ...
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0answers
29 views

Definition of hypersurface singularity

I am really confused about this notion. Suppose $X$ is an arbitrary variety over an algebraically closed field $k$ (if you like, let the characteristic be $0$), and $p$ is a $k$-valued point. If $p$ ...
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1answer
50 views

What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...
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0answers
33 views

Help to understand the proof of the Riemann Mumford relation

Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13. My problem is to understand the beginning of that proof. In particular ...
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1answer
36 views

Page 276 of Principles of Algebraic Geometry by Griffiths and Harris; wrong parameter count?

The following is taken from page $276$ of Principles of Algebraic Geometry by Griffiths and Harris: Now let $S$ be a Riemann surface of genus $g\ge 3$. By our last result, if $S$ has any ...
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1answer
41 views

Question about the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$ in Principles of Algebraic Geometry by Griffiths and Harris.

On page $115$ of Principles of Algebraic Geometry by Griffiths and Harris, in the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$, they state that ...
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115 views
+50

Complex manifold with subvarieties but no submanifolds

There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. For example, generic tori of dimension greater than one have no compact complex submanifolds. ...
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0answers
26 views

if $F_{\bullet}$ is a complex and $r$ an integer, what is $F_{r-\bullet}$?

While reading the paper Some results and questions on the Castelnuovo-Mumford regularity, by Marc Chardin, I encountered in the proof of Theorem 5.1 the notation $F^N_{r-\bullet}$. To provide some ...
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1answer
144 views

Is there a (not so) generalized version of Hilbert's Theorem 90?

I'm sorry if my following question doesn't make any sense. We know that if $L/k$ is a finite Galois extension then $H^{1}(\mathrm{Gal}(L/k),L^{*})=0$ (Hilbert's theorem 90). However I would like to ...
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0answers
31 views

Recommend guide book of algebraic geometry [duplicate]

I have a little knowledge about geometry and algebraic topology . I want to learn some basic conception and thought of algebraic geometry. Besides , I want to know main of theory of sheaves. What book ...
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0answers
29 views

Finding Irreducible components

I want to find the irreducible components of variety $V=(\langle x^3-x, xy^2+2z^2-x\rangle) \subset \mathbb{A}^3(\mathbb{C})$. But, it doesn't seem to be straight forward since I have two polynomials ...
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1answer
18 views

Algebraic curve locally diffeomorphic to its tangent space at a regular/smooth point.

Let $\sf C$ be an equidimensional algebraic curve of $\mathbb{C}^n$. Let $x$ be a point of $\sf C$ witch is a regular/smooth point. I want to prove that there exists a Zariski open set $O$ such that ...
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0answers
31 views

Toposes in algebraic geometry

I know the definition of topos and read an introduction to algebraic geometries. I heard that topos is used for algebraic geometry and want to know detail. Where can I read about this?
1
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1answer
52 views

Problem on co-ordinate geometry

Suppose the circle with equation $x^2 + y^2 + 2fx + 2gy + c = 0$ cuts the parabola $y^2 = 4ax$, ($a > 0$) at four distinct points. If d denotes the sum of ordinates of these four points, then find ...
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33 views

Irreducible variety

I have the following problem and seems to stuck with some basic understanding of irreducible and/or non-singular varieties. In $\mathbb{P}^3$ we have an irreducible variety $A$ given by two equations ...
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0answers
15 views

Constructing Incidence variety without using equations

Let $k$ be a field. Let $X$ be the Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a specified Hilbert polynomial. Let $Y$ be another Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a ...
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1answer
15 views

Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with any line l is at most 2.

I came across this question in Algebraic Geometry: A Problem Solving Approach: Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with ...
2
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0answers
68 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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1answer
34 views

If a linear transformation is defined over $F$, so is the kernel

Let $V$ be a vector space over a field $k$, and $F$ a subfield of $k$. An $F$-submodule $V_0$ of $V$ is called an $F$-structure if the natural $k$-linear map $V_0 \otimes_F k \rightarrow V$ is an ...
0
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1answer
37 views

Cohomology groups of a non-degenerate algebraic variety.

Let $X\subset\mathbb{P}^{n}$ be an algebraic variety. Let us suppose that $X$ is non-degenerate (it is not contained in any hyperplane of $\mathbb{P}^{n}$). I have read that (at least for curves) the ...
2
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0answers
36 views

Changes of Cohomologies in small resolution

Let $X$ be a singular complex variety of dimension $3$, whose singular locus is only a node! Suppose there exists a small resolution \begin{equation} \pi~:~\hat{X} \rightarrow X \end{equation} which ...
3
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1answer
18 views

A question about Weil restriction

Let $l/k$ be a finite Galois extension of fields of characteristic zero. Let $X$ be an affine scheme of finite type over $l$ and denote the Weil restriction by $\prod_{l/k} X$ (it exists in this ...
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1answer
29 views

Relative Frobenius Morphism of Finite Fields

Let $K$ be a finite field of characteristic $p$ and let $L$ be a finite extension of $K$. Then $L$ has an absolute Frobenius morphism which is given by the $p$th power map. Moreover, we have a map of ...
2
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1answer
32 views

Arithmetically Cohen-Macaulay curve on a quadric

If $Y$ is a curve of bidegree $(a,b)$ on a smooth quadric surface $Q\subset \mathbb{P}^3$, how do we see that it is arithmetically Cohen-Macaulay (ACM, for short) iff $|a-b|\leq 1$? If (like me) ...
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1answer
38 views

Finding Singular points

Let $f = 3x^3 + 3x^2 − y^2 + z^2$, $g = 3x^2 + 4^x + 3y^2 + z^2$ be polynomials in $\mathbb{C}[x, y, z]$ and let $W = V(\langle f, g\rangle) ⊂ \mathbb{A}^3(C)$. By using the Jacobian matrix, find the ...
2
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1answer
68 views

Structure sheaf of affine variety consists of noetherian rings

Let $X\subseteq \mathbb{A}^n$ be an affine variety. The local ring of $X$ at $p\in X$, given by $\mathcal{O}_{X,p}=\{f\in k(X):f \text{ regular at } p\}$ is noetherian because it is a localization of ...
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0answers
26 views

whats wrong with this counterexample to closed subgroups of a Torus are a torus

In Cox Little and Schenck, one result that is cited in chapter two is that if $D_n$ is the $n-dimensional$ torus, and $H < D_n$ is a closed subgroup then $H$ is itself a torus. Let the underlying ...
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34 views

Finite type assumption necessary for this property of very ample sheaves?

The question $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample asks for a proof of the following statement: ...
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23 views

Can you help me with some elementary articles about the link between regular points of algebraic sets and regular local rings? (With many examples) [closed]

Can you help me with some elementary articles about the link between regular points of algebraic sets and regular local rings? (With many examples)
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8 views

Four polytopes and their relation

Suppose we have four similar $A_{1},A_{2},A_{3},A_{4}$ polytopes in Euclidean Space. They are different and we know that $$ A_{1}\cap A_{2}=B_{1},~A_{2}\cap A_{3}=B_{2},~A_{3}\cap ...
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1answer
19 views

Weirestrass points in Principles of Algebraic Geometry.

On page 274 we have the gap values of $p\in S$ where $S$ is a Riemann surface, these are listed as follows: $$a_1 = 1 , a_2 = 2+\alpha_1 , \ldots , a_g = g+ \alpha_1 + \ldots \alpha_{g-1}$$ Now the ...
3
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1answer
37 views

Computing dimension of a smooth scheme

I'm having trouble finding enough reference to show the following seemingly true statement. Let $K$ be a local field with ring of integer $\mathcal{O}_K$. Assume that $X$ is a smooth ...
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1answer
35 views

On the proof that one dimensional linear algebraic groups are either isomorphic to $\mathbb{G}_m$ or $\mathbb{G}_a$.

Let $G$ be a linear algebraic group of dimension one. The proof that I am looking at, in t.a springer's book (thm 3.4.9) proceeds by showing that $G$ must be either equal to its semisimple part ...
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0answers
61 views

My strange proof of the fact that $k[\mathbb P^n]=k$

Let $f\in k[\mathbb P^n]$, i.e. $f: \mathbb P^n \to \mathbb A^1$ be a regular function. My purpose is to show $f$ is constant. First $\mathbb A^1$is considered the subset $\{x=[x_0:x_1]\in \mathbb ...
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0answers
44 views

The sheaf (of stalks) of meromorphic functions, why don't we use a more natural definition?

If $A$ is a commutative ring with $1$, let's denote with $R(A)$ the set of regular elements of $A$. Let $(X,\mathcal O_X)$ be a locally Noetherian scheme, then the sheaf (of stalks) of meromorphic ...
2
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1answer
29 views

Embedding Complex Tori in Projective Space

When we talk about projectively embedding complex tori $\mathbb{C}^{g}/\Lambda$ (i.e in Lefshetz Embedding Theorem), what exactly do we mean by an embedding. Is it in the differential geometry sense ...