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

Exact sequence of sheaves and associated sequence of graded modules

Let $(X,\mathcal{O}_X)$ with $X=\mathbb{P}^n$ and consider a exact sequence of sheaves of $\mathcal{O}_X$-modules $$0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0 $$ Suppose that we apply the ...
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29 views

Twisting sheaf is invertible.

I have a small question in the proof of Hartshorne's book of the fact that $\mathcal{O}(1)$ is locally free. The thing is that it suffices to prove that $$ \mathcal{O}(1)(D^{+}(f)) \cong ...
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24 views

A Plucker style proof of Monge's Theorem

Plucker, famously, proved Pascal's theorem for all conics at once, using the technique described in the answer here. I was wondering if there was a proof for Monge's Theorem using the above ...
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2answers
89 views

Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
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54 views

morphism of sheaves on $\mathbb{R}/\mathbb{Z}$

Let $\mathscr{Z}$ be an arbitrary sheaf on $\mathbb{R}/\mathbb{Z}=X$ (with the quotient topology). Let $\mathscr{F}$ and $\mathscr{G}$ denote the sheaves of continuous functions on $X$ with values in ...
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30 views

Local ring of an affine curve $K$ at a point $p\in K$

I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows: The local ring of $K$ at $P=(a,b)$ is the ring ...
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38 views

Cartier divisor and map to associated line bundle

Given a Cartier divisor $D$ on an integral, separated scheme $X$ of finite type over an algebrically closed field $k$. Does such a divisor always induce a map $\mathcal{O}_X \to \mathcal O_X(D)$? I ...
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1answer
40 views

Example of an irreducible algebraic set consisting of more than one polynomial

By definition, an algebraic set is a zero locus of polynomials: $$ \{x\in \mathbb A^n \mid p(x) = 0 \,\,\,\, \forall p \in S\}$$ where $S$ is a set of polynomials $p \in k[x_1, \dots, x_n]$. It is ...
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60 views

Ideal of 8 general points in $\mathbb{P}^2$

I am working through chapter 3 of Eisenbud's Geometry of Syzygies. In the first example he makes the claim that the ideal of 8 general points in $\mathbb{P}^2$ is generated by two cubics and a ...
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206 views

How is this fact implicated?

Let $K$ be a field and $\overline{K}$ its algebraic closure, then we define the $n$-dimensional affine space as $$\mathbb{A}^{n}=\{(x_1, \ldots, x_{n})\mid x_1, \ldots, x_{n} \in \overline{K}\}$$ So ...
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37 views

What is the stalk of the structure sheaf of the plane?

Let $\mathcal{O}$ be the structure sheaf of $\mathbb{A}^2_\mathbb{C}$. How do I compute the local ring corresponding to the stalk of $\mathcal{O}$ of the point $(0,0)$? I tried computing the ...
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1answer
42 views

Isomorphic surfaces in $\mathbb{P}^3$

If $X_0,X_1\subset\mathbb{P}^3$ are surfaces of degree $d\geq 5$ that are isomorphic as abstract surfaces, why is there an automorphism of $\mathbb{P}^3$ that induces an isomorphism between $X_0$ and ...
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53 views

Which polynomials are resultants?

Let $f(x,y),g(x,y)\in\mathbb{Q}[x,y]$ with degrees $\deg(f)=m,\deg(g)=n$. Considering these polynomials as univariate polynomials in $y$ over the field $\mathbb{Q}[x]$, the resultant ...
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1answer
20 views

'Trivial' embeddings have the same degree?

We can define the degree of a projective variety $X\subseteq\mathbb{P}^n$ in terms of the maximal number of intersections with projectivisations $L=\mathbb{P}(\hat{L})$ of linear varieties ...
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52 views

Hilbert polynomial of iterated Veronese embedding

Let $X=\mathbb{V}(x^2-yz)\subset\mathbb{P}^2$ and consider the Veronese embedding $Y=\mathcal{v}_2(X)\subset\mathbb{P}^5$. Find the Hilbert polynomial, and thus the degree, of $Y$. I know how we can ...
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1answer
44 views

Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
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43 views

Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
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12 views

Finding Primitive Elements of Separable Function Field Extensions

Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields ...
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71 views

History and future of algebraic curves and the like?

Now that Fermat's last theorem has been proven, and also elliptic curves see widespread use in simple everyday applications, I would love to learn how the related theories came into beeing, how they ...
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1answer
30 views

Action of the group of automorphisms of a connected finite étale cover (Corollary 5.3.4 in Szamuely).

I have difficulty to understand the proof of Corollary 5.3.4 in Szamuely, Galois group and fundamental groups. The proof use the following corollary. Corollary 5.3.3. If $Z \longrightarrow S$ is a ...
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23 views

Dimension of intersection of affine varieties

Let $X,Y \subset \mathbb{A}^n$ be two irreducible affine varieties with nonempty intersection. Prove that dim$X \cap Y \geq$dim$X+$dim$Y-n$ There is a hint to use the diagonal $\Delta=\{(x,x)|x \in ...
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1answer
30 views

Local argument for proving that $\operatorname{Ext}^1$ vanishes for two sheaves

I am trying to compute the vanishing of $\operatorname{Ext}^1$ for two sheaves of $\mathcal{O}_X$-Modules and I was wondering if it was possible to use some local argument to reduce the problem to ...
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1answer
41 views

Push forward of an exact sequence of sheaves under blow up

Let $X$ be a smooth projective variety of dimension $n\geq 3$. Let $Z$ be a smooth subvariety of $X$ of codimension at least 3. Let $Y$ be the blow up of $X$ along $Z$ and let $f:Y\longrightarrow X$ ...
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24 views

How many points at infinity in Artin-Schreier type curve

Let $Y$ be an affine curve over a perfect (yet not necessarily algebraically closed) field $k$ given by $$y^p+a(x)y=b(x)$$ (abs. irreducible) with $p$ a prime number. Now one can normalize $k[1/x]$ in ...
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1answer
19 views

Regular functions extension to normal points of varieties

I am doing the exercise 3.20 in Robin Hartshorne's Algebraic Geometry, Chapter 1. Let $Y$ be a variety of dimension $\geq2$, and let $P\in Y$ be a normal point. Let $f$ be a regular function on ...
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1answer
38 views

Example of a projective variety that is not projectively normal but normal

I want to prove the following statement: Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal. ...
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1answer
23 views

Finding algebraic curve satisfying given parameterization

Is there an easy way to find an algebraic curve that satisfies a given parameterization? Specifically, I am talking about the following parameterization: $$ x=z(1-z),\hspace{10pt} y=\sum_{n=1}^r ...
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1answer
29 views

$Z(y^2-x^3) \subset \mathbb{A}_{\mathbb{R}}^2$ is not isomorphic to $\mathbb{A}_{\mathbb{R}}^1$

Prove that the algebraic variety $Z(y^2-x^3) = \{(x, y)\in\mathbb{A}_{\mathbb{R}}^2\,\,|\,\,y^2-x^3=0\}$ is not isomorphic to the affine space $\mathbb{A}_{\mathbb{R}}^1$. [i.e., there are no ...
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62 views

About alternative ways of computing $H^1(X,\mathcal{O}_{\mathbb{P}^n}(m))$.

This is a follow up to my question : Applications of $Ext^n$ in algebraic geometry In the case of $\mathcal{O}_X$-Modules it is clear that $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ ...
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1answer
31 views

Affine Zariski topology is normal

Let $C,D$ be two disjoint Zariski-closed subsets of $\mathbb{C}^n$, and let $f,g$ be polynomial functions on $C,D$ correspondingly. Then there is a polynomial function $h$ on $\mathbb{C}^n$ that ...
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1answer
40 views

Foliation dense if $G = \textbf{R}$, where $G$ is a subgroup of a Lie group $G'$.

I have the following statement: Let $G$ be a subgroup of a lie group $G'$, and the action is left multiplication. The leaves are then the left cosets of $G$ in $G'$. If for example, we let $G = ...
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34 views

Is this criterion for projective normality missing a hypothesis?

I proved the following: Let $S=S(X)$ be the homogeneous coordinate ring of a connected, normal closed subscheme of $\mathbf P^r_A$, where $A$ is a ring. Then $S$ is a domain, and ...
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18 views

Moving from polynomials into radical functions: does algebraic geometry still work?

Say I have a polynomial: $x + y = 0$, but subject to the constraints $x = \sqrt{1-u^2}$, $y = \sqrt{1-w^2}$, $u\in[-1,1],w\in[-1,1]$. I can re-write my original polynomial as an inequality: ...
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1answer
22 views

Dimension of cone of projective variety

Let $X \subset \mathbb{P}^n$ be a nonempty projective variety. Show that the dimension of the cone $C(X):=\{0\} \cup \{(x_0,...,x_n)\in \mathbb{A}^{n+1}:(x_0:...:x_n)\in X\}$ is dim$X+1$. I know how ...
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1answer
20 views

Finding solution of a system of polynomials through intersecting sub solutions?

I have a conjecture but I'm not sure if it's true. Intuitively, it seems correct, but... here it is: Conjecture: Let $S$ be a set of polynomials in $n$ variables over $\mathbb{C}^n$ (*see below). ...
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22 views

Deformation of a point on a Quot scheme

Let $\mathcal{H}$ be a coherent sheaf on a projective variety $X$. We say that a sheaf $\mathcal{E}$ is of $\textit{pure dimension}$ if for all non-trivial coherent subsheaves $\mathcal{E'} \subset ...
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18 views

Singular affine monomial Curve.

To me, an affine monomial curve $C$ is parametrized as $C:(t^{m_1},\cdots,t^{m_r})$ with $\textrm{G.C.D.}\{m_1,\cdots,m_r\}=1$ and $m_1<\cdots<m_r$. How show that $C$ is singular in ...
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1answer
37 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on ...
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2answers
88 views

Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$ [closed]

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.
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25 views

Computing Intersection Product in a Cell Complex

Here's the information I have: I have an abstract cell complex that represents some space I am studying, and it is known that the space is orientable. I can identify the sub complexes which ...
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1answer
31 views

Correspondence between prime ideals and irreducible algebraic sets

Let $k$ be an algebraic closed field. The Nullstellensatz theorem prove that $$I(V(J))=\sqrt{J}$$ and we have $$V(J)\text{ irreducible }\iff I(V(J)) \text{ prime }$$ So if $J$ is prime, $I(V(J))=J$ is ...
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14 views

The Variety X(K) as a subset of the analytification of X

I am working with Sam Payne's artical Analytification is the limit of all tropicalizations, and because of my limited understanding of analytification it gives me some difficulties. The article: ...
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54 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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2answers
38 views

Dimension of irreducible affine variety is same as any open subset

Let $X$ be an irreducible affine variety. Let $U \subset X$ be a nonempty open subset. Show that dim $U=$ dim $X$. Since $U \subset X$, dim $U \leq$ dim $X$ is immediate. I also know that the result ...
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1answer
42 views

What is the ramification locus of $Spec\mathbb{Z}[x]\rightarrow Spec\mathbb{Z}[x]$ given by $x\mapsto x^2$?

$\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\Spec}{\text{Spec }}$ Let $A = B = \ZZ[x]$, and consider the map $B\rightarrow A$ given by $x\mapsto x^2$. Intuitively, $f : \Spec A\rightarrow\Spec B$ ...
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30 views

Non-proper intersections and singular points

Let $X$ be a smooth variety in $\mathbb{P^N}$. Consider an intersection $Z=X\cap\mathbb{P}^n\subset\mathbb{P}^N$ with some subspace $\mathbb{P}^n$ and assume that $\dim Z>\dim X+n-N$. Is it true ...
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87 views

How to tell if a system of polynomial equations has no real solutions

I have a system of $3n + 3$ polynomial equations in $6n$ variables, where $n$ is probably going to be less than about $5$. I can compute its Groebner basis and I see that it does not contain $\{1\}$, ...
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39 views

Subspace of topological space has lower dimension

The dimension of a topological space $X$ is defined to be the length of the maximal chain of closed irreducible subsets $\varnothing \neq X_0 \subsetneq X_1 \subsetneq ... \subsetneq X_n \subset X$. ...
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1answer
14 views

Space admitting an irreducible connected open covering is irreducible

Let $\{U_i:i \in I\}$ be an open covering of topological space $X$, where $U_i \cap U_j \neq \varnothing$ for every $i,j$. If $U_i$ is irreducible for all $i \in I$ then $X$ is irreducible. I am ...
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23 views

Constructing an algebraic surface with singularities on the unit circle.

I am currently doing a project on algebraic surfaces, and I want to construct an algebraic surface $\mathbb{V}(f(x, y, z))$ that exhibit its singularities on the unit circle $x^2 + y^2 - 1 = 0$. My ...