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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1answer
196 views

What is generic coordinates?

in the book Irena Peeva "Graded Syzygies", I saw that Polynomial ring has generic coordinates. But in this book, It is not defined anywhere. I looked for it but did not see a clear definition. I want ...
2
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0answers
381 views

Recovering the topology of an affine scheme from the specialization preorder

Let $A$ be a commutative ring. The specialization preorder on $\mathrm{Spec}(R)$ is given by $\mathfrak{p} \prec \mathfrak{q} \Leftrightarrow \mathfrak{p} \in \overline{\{\mathfrak{q}\}} ...
4
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0answers
170 views

Definition of stalk as a colimit of sheaves

I'm trying to understand the category-theoretic proof that sheafification preserves stalks, using adjoints, as outlined e.g. in this mathoverflow answer: ...
4
votes
1answer
145 views

Higher direct image and local cohomology.

Let $X$ be an scheme, $Z \subset X$ a closed subscheme, and $\mathcal{F}$ a coherent sheaf then, $\mathcal{R}^{i-1}_{j_{*}}(\mathcal{F}|_{X-Z})\cong\mathcal{H}_{Z}^{i}(X,\mathcal{F})$ I would like ...
10
votes
3answers
250 views

Is every algebraic curve birational to a planar curve

Let $X$ be an algebraic curve over an algebraically closed field $k$. Does there exist a polynomial $f\in k[x,y]$ such that $X$ is birational to the curve $\{f(x,y)=0\}$? I think I can prove this ...
6
votes
1answer
127 views

Separated schemes and unicity of extension

In point set topology, we have the following result, which is easily proved. Theorem. Let $Y$ be Hausdorff space and $f,g:X \to Y$ be continuous functions. If there exists a set $A\subset X$ such ...
2
votes
2answers
126 views

Embedding elliptic curves into the general linear group

Is it possible to embedd an elliptic curve $E:\;\; y^2=x^3+ax+b$, defined over an algebraically closed field $k$, into some $GL_n(k)$ ?
2
votes
1answer
182 views

Coherent Sheaves on Projective Space

I am having trouble proving the following claim and would be glad if someone could help me out. Claim: Let $\mathbb P$ denote n-dimensional projective space, and let $F$ be a coherent sheaf on ...
2
votes
1answer
283 views

Two notions of uniformizer

Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
2
votes
1answer
241 views

$GL_n(k)$ (General linear group over a algebraically closed field) as a affine variety?

In the context of linar algebraic groups, I read in my notes from the lecture that's already some while ago that $GL_n(k)$ is an algebraic variety because $GL_n=D(\det)$, $ \det \in k [ (X_{ij})_{i,j} ...
2
votes
1answer
229 views

Hyper-elliptic curves in positive characteristic

I have been looking at hyperelliptic curves in characteristic two, in particular using Algebraic Geometry and Arithmetic Curves by Qing Liu, which gives a description in all characteristics. For the ...
4
votes
2answers
232 views

When does a morphism preserve the degree of curves?

Suppose $X \subset \mathbb{P}_k^n$ is a smooth, projective curve over an algebraically closed field $k$ of degree $d$ . In this case, degree of $X$ is defined as the leading coefficient of $P_X$, ...
5
votes
1answer
348 views

Hilbert's Nullstellensatz without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. Can we prove the following theorem without Axiom of Choice? Theorem Let $A$ be a commutative ...
9
votes
1answer
202 views

Any affine algebraic group is linear.

It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
1
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0answers
84 views

base change and constructible sheaves

let $f:X\to S$ be a proper morphism of schemes with $S=Spec(A)$ affine. Consider $F$ a constructible sheaf on $X$. I am interested to know for which ring $B$ with morphism $Spec(B)\to Spec(A)$ is it ...
4
votes
1answer
399 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
5
votes
1answer
441 views

When a scheme theoretical fiber is reduced?

I'd like to ask some basic things in algebraic geometry. Suppose I have a map $\phi:V\to W$, between affine varieties over $k=\mathbb{C}$. for any point $y \in W$. The scheme theoretical fiber is ...
6
votes
1answer
117 views

Trivial Restriction of Line Bundles

Say I have some projective space $\mathbb{P}^n$ and some line bundle $L=\mathcal{O}(-k)$. Now, I want to have a subvariety $Y$ in $\mathbb{P}^n$ such that $L\vert_Y$ is trivial. When is this the ...
2
votes
1answer
252 views

From Presheaf to Sheaf

In Hartshorne's Algebraic Geometry is written that "A sheaf is roughly speaking a presheaf whose sections (i.e. elements of $\mathcal{F}(U)$ for open subset $U$) are determined by local data". What ...
0
votes
1answer
152 views

Smallest genus example of a non planar curve

A curve is a smooth projective connected curve over an algebraically closed field. Every curve of genus 2 is planar. Also, every curve of genus 3 is planar. But what about curves of genus 4? What ...
8
votes
0answers
169 views

Tensoring is thought as both restricting and extending?

I hope these questions are not too trivial. Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring $$ (R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c ...
0
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0answers
58 views

About stalks of a sheaf. [duplicate]

Possible Duplicate: Failure of isomorphisms on stalks to arise from an isomorphism of sheaves Let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over a topological space X. If we have ...
5
votes
1answer
597 views

Chern Class = Degree of Divisor?

Is the first chern class the same as the degree of the Divisor? Say, $C$ is some divisor on $M$, is $c_1(\mathcal O (C)) = \text{deg }C$? And say I have some Divisor $D$ with first chern class ...
4
votes
1answer
146 views

If the special fiber of a flat morphism is reduced, then any other fiber is reduced?

Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...
5
votes
1answer
271 views

trivial Picard group

let $S=\operatorname{Spec}(A)$ be an affine scheme. For which ring $A$, not field is it known that $H^1(S,\mathcal{O}_S^{*})$ is trivial? If $X\to S$ is a finite map and $H^1(S,\mathcal{O}_S^{*})$ is ...
9
votes
3answers
894 views

Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
2
votes
1answer
190 views

Congruence subgroups and modular curves of type (M,N)

I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$). Let $\Gamma(M,N)$ be ...
1
vote
1answer
389 views

relating flatness, equidimensional, and complete intersection

I am a bit confused and am trying to clarify some notions. First consider the following well-known statement. A dominant map $f:X\rightarrow Y$ between regular varieties is flat if and only if it is ...
2
votes
1answer
80 views

Extension of morphisms on surfaces

Consider two regular integral proper algebraic surfaces $X$ and $Y$ over a DVR $\mathcal O_K$ with residue field $k$. Let $U \subset X$ be an open subset, s.t. $X\setminus U$ consists of finitely many ...
1
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0answers
71 views

Partial fraction expansion of rational functions

I've been trying to read about partial fraction expansion of rational function. Is the following statement equivalent to the uniqueness+existence of partial function expansion?: Let $\mathbb{F}$ be ...
4
votes
0answers
81 views

the existence of a closed subset of $\mathbb{A}^8\times \mathbb{A}^1$ which is flat over $\mathbb{A}^1$

I've been doing some reading on deformation theory and one way it is used is to study singularities on varieties while perturbing the varieties. I would actually like to use deformation theory to ...
2
votes
1answer
155 views

Divisor of degree 2 on a smooth plane curve

Let $X$ be a smooth plane curve of genus $3$ (assume a smooth plane quartic) and $D$ a divisor of degree $2$ on this curve. Assume that $\mathcal{l}(D)>0$. It means that there exists a rational ...
3
votes
1answer
109 views

How to handle group schemes by points?

I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. ...
0
votes
1answer
49 views

Reducing a flat morphism $\psi:X\rightarrow\mathbb{A}_{\mathbb{C}}^1\;$ to $\;\psi|_{Y}: X\cap Y\rightarrow \mathbb{A}_{\mathbb{C}}^1$

Suppose $\psi: X\rightarrow \mathbb{A}_{\mathbb{C}}^1$ is a flat morphism, where $X\subseteq \mathbb{A}_{\mathbb{C}}^n$ with $X$ not needing to be smooth, with $\psi^{-1}(0)$ being a complete ...
0
votes
1answer
77 views

explicitly constructing a certain flat family

Is it possible to construct a flat family $$ \phi:\mathbb{A}_{\mathbb{C}}^8=\operatorname{Spec} \mathbb{C}[x,y,z,w,a,b,c,d]\longrightarrow \operatorname{Spec} \mathbb{C}[t_1, t_2, t_3] ...
0
votes
0answers
1k views

Converting standard equation for a paraboloid to a parametric one

I have the equation for a hyperbolic paraboloid in $x$, $y$, and $z$: $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$ I also have the parametric equations for the same parabaloid: $$x = a u ...
2
votes
0answers
59 views

Descent through blow-up

Let $X$ be a variety with $Y \subsetneq X$ a proper closed subvariety. Let $Z$ denote the blow-up of $X$ along $Y$. Let $f: Z \rightarrow X$ be the canonical map. Suppose that we have a coherent sheaf ...
2
votes
1answer
82 views

If $\phi^{-1}(0)$ in $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a complete intersection, then is $\phi$ flat?

This is a simple question so I am hoping the answer is quite simple as well. Suppose $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a map such that the algebraic variety $\phi^{-1}(0)$ is a complete ...
4
votes
2answers
157 views

zeroes of forms on Riemann surfaces

Let $P$ be a point on a Riemann surface. Does there exist a non-trivial differential form $\omega$ on $X$ such that $\omega$ vanishes at $P$? Does there exist a non-constant rational function $f$ on ...
1
vote
2answers
109 views

Regular functions on $\mathbb P_k^n$

Let be $k$ an algebraically closed field and let's consider a projective algebraic set $V\subseteq\mathbb P^n_k$ with the induced Zariski topology. If $U\subseteq V$ is open, likewise the affine ...
3
votes
0answers
223 views

There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
6
votes
2answers
195 views

Actually calculating an Intersection of Variety and Divisor

This has been bugging me for a while now. Say I have a projective variety given by some polynomial $P$ and the canonical divisor of the projective space. How can I concretly calculate the ...
0
votes
1answer
137 views

References for Vector bundle over a projective space?

I know just the basics of sheaf theory and would like to ask about good references for "vector bundle over a projective spaces"?
3
votes
1answer
71 views

Separatedness of a composition, where one morphism is surjective and universally closed.

I'm stuck with the following problem: Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be scheme morphisms such that f is surjective and universally closed and such that $g \circ f$ is separated. The ...
1
vote
2answers
111 views

quotients of curves by actions of roots of unity

let $X$ be a smooth projective irreducible curve of genus $g$ over the complex numbers. Assume that $X$ comes with an action of $\mu_d$. Is the quotient $Y:=X/\mu_d$ always smooth? Let $\pi: X \to ...
1
vote
1answer
237 views

Sheaf of rings with vanishing stalk?

How common is that a sheaf of rings has a vanishing stalk? To define the rank of a locally free sheaf of $\mathscr{O}$-modules, for instance, $\mathscr{O}_x=0$ may cause some problem, since the rank ...
5
votes
1answer
237 views

Taking stalk of a product of sheaves

Let $(\mathscr{F}_\alpha)_\alpha$ be a family of sheaves on $X$, and $\prod_\alpha\mathscr{F}_\alpha$ the product sheaf. If $x\in X$, is it true that ...
3
votes
1answer
305 views

Is locally free sheaf of finite rank coherent?

Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent? By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that ...
4
votes
1answer
146 views

Cech cohomology of $\mathbb A^2_k\setminus\{0\}$

I'm trying to prove, via the Cech cohomology, that $S=\mathbb A^2_k\setminus\{0\}$ with the induced Zariski topology is not an affine variety. Consider the structure sheaf $\mathcal O_{\mathbb ...
4
votes
1answer
220 views

codimension of “jumping” of the dimension of fibers

Let $f:X\rightarrow Y$ be a dominant morphism of projective (and smooth if you like) varieties over an algebraically closed field $k$ such that $n=\dim(X)=\dim(Y)$. Then $f$ is proper, so by ...