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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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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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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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
38 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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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
41 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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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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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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27 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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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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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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32 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?
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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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2answers
83 views

Why are Del Pezzo surfaces rational?

Let $X$ be a Del Pezzo surface over an algebraically closed field $k$, i.e. a projective surface with $-K_X$ ample. I'm chiefly interested in the case that $k$ has characteristic $0$ and $X$ is ...
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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
17 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
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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1answer
16 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 ...
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Question regarding Geometric meaning of Noether normalization theorem for projective varieties

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states: Let $K$ be an algebraically closed field, $V\subset \mathbb{P}^n(K)$ a variety of ...
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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 ...
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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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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 ...
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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 ...
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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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1answer
33 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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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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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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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 ...
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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 ...
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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 ...
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62 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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45 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 ...
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50 views

When a prime ideal is maximal differential ideal in a UFD?

Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with derivatives $D(X)=Y, D(Y)= -X$? I know there are maximal ideals like ...
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1answer
45 views

Tensor product of coordinate rings corresponds to pullback

Here in Milne's notes on algebraic geometry, he proves that if $k$ is an algebraically closed field, and $A$ and $B$ are reduced finitely generated $k$ algebras, then $A \otimes_k B$ is reduced. (This ...
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37 views

Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
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1answer
36 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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55 views

Why are there no $\mathbb{R}$-valued points on a complex curve?

We say a $K$-valued point on a scheme is a map Spec$(K) \to S$, so in particular, a real valued point on the parabola $y = x^2$ should be a map Spec$(\mathbb{R}) \to $Spec$(\mathbb{C}[x,y]/(y-x^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 ...
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A characterization of irreducible polynomials linear in one variable?

Let $p\in \mathbb{C}[x,y,z]$ be an irreducible polynomial such that for each $x,y\in (−1,1)$, there exists a unique $z_{x,y}\in \mathbb{C}$ suchthat $p(x,y,z_{x,y})=0$. Conjecture: There exist ...
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Differential of the Gauss map of an algebraic variety.

Let $X=V(F)\subset\mathbb{P}^{n}$ be a smooth irreducible hypersurface. Let us consider the morphism $$ \mathcal{G}:X\rightarrow \mathbb{P}^{N}, p\mapsto \left( \frac{\partial F}{\partial ...
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1answer
57 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 ...
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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
34 views

Show that $D$ is not linearly equivalent to any other effective divisor

Let $C$ be a nonsingular quartic, $P_1,P_2,P_3 \in C$. Let $D=P_1+P_2P_3.$ Let $L$ and $L'$ be lines such that $L \bullet C= P_1+P_2+P_4+P_5$ and $L'\bullet C= P_1+P_3+P_6+P_7.$ Suppose these seven ...
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1answer
47 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ...
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1answer
30 views

Moving Line Segment Problem part 2

This question is related to a question I asked a while ago here on math.stackexchange: Moving Line Segment Problem The rules for how the line segment can be moved are the same: The endpoints must ...
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1answer
77 views

Luna-Vust theory for embeddings of homogenous spaces

I'm interested in the theory of Luna and Vust of embeddings of homogenous spaces like presented in D. Luna, Th. Vust: Plongements d'espaces homogènes, Comment. Math. Helvetici 58 (1983) 186-245. ...
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1answer
28 views

Cartier divisor and its support.

(1) Let $X$ be a Noetherian scheme of dimension 1 over a field $k$ and $supp(D)$ denote the support of a $Cartier$ divisor $D$ on $X$. Let $S\subseteq supp(D)$ be consists of closed points of ...
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1answer
25 views

Quotient Sheaves

Let $X$ be a ringed space, and $J$ be a sheaf of ideals of the structure sheaf. Define, $Y = \{x\in X ~ | ~ J_x \not = \mathcal{O}_x\}$, this is a closed set. We have an inclusion $i:Y\to X$. Is there ...
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1answer
35 views

Schemes not determined by morphisms from one object

I was reading a bit about the functor of points of a scheme, and it was mentioned that there does not exist a scheme $Y$ so that Hom$(Y,X)$ determines the points $X$ for all schemes $X$. This is in ...