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
56 views

Smoothness for morphism of schemes

Let $X \to Y$ be a projective morphism of schemes of finite type with $Y = Spec(R)$, where $R$ is a dvr. For this morphism to be smooth, is it sufficient to check smoothness on only closed points of ...
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13 views

Dirrefential of boundary morphisms in the moduli space of pointed stable curves.

Recall that first order deformations of a smooth pointed curve $(C,p_1,\ldots,p_n)$ are parametrized by $H^1(C,\cal{T}_C(-p_1-\ldots-p_n))$ and in the stable case is ...
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1answer
35 views

Pushforward and sheaf-hom

If $S$ is a surface over the complex numbers $\mathbb{C}$, and $C$ is a curve in $S$, and $i:C\longrightarrow S$ is the inclusion morphism. If $A$ is a line bundle over $C$, then is it true that ...
5
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1answer
92 views

Complement of open set is finite in Zariski topology

This problem has two parts: a) Let $M$ be a finitely generated module over a Noetherian ring $A$. Prove that $S=\{ P \in\operatorname{Spec}(A) : M_P \mbox{ is a free }A_P\mbox{-module} \}$ is an ...
4
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1answer
129 views

example of locally finite type not finite type

Let $A$ be a ring then, a homomorphism $A\rightarrow A[x_1,\cdots,x_n]$ induces a finite type morphism between spectrums. I want to find the map which is a locally finite type but not finite type.... ...
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49 views

Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
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32 views

morphisms of curves and discrete valuation rings

Given a dominant morphism $\varphi\colon C\to C'$ of curves, a nonsingular point $Q\in C'$, such that $\varphi^{-1}(Q) = \{P_1,\ldots, P_m\}$ consists of nonsingular points only. Then it is clear to ...
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1answer
39 views

Flat Module finitely generated when over the residue field finite dimensional? [on hold]

Let $(A, \mathfrak{m})$ be a local ring with residue field $\kappa=A/ \mathfrak{m}$. Let $M$ be a flat $A$-module. Assume that $M \otimes_A \kappa$ is a finite dimensional $\kappa$-vector space. Is it ...
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1answer
39 views

If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?

Suppose you have a pair of integral varieties $X$ and $Y$ such that $\dim(X)=\dim(Y)>0$, and there exists a surjective morphism $X\to Y$ between them. I was wondering, if $X$ is affine, does this ...
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49 views

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(X)$ generated by $\sqrt{X^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
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1answer
60 views

Construction of a line bundle with lower degree and lower dimension of global sections

Let $\mathcal{L}$ be a line bundle on a smooth projective irreducible curve $C/k=\overline{k}$ with genus $g$ and $\text{deg}\mathcal{L}=g$. Assume $\text{dim}H^0(C,\mathcal{L})>1$. It's my aim ...
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41 views

Dimension of irreducible components of variety [on hold]

Consider the affine variety $X=\{ (a_1,a_2,a_3,b_1,b_2,b_3) \in \mathbb{C}^6 \mbox{ : }a_1b_2=a_2b_1, a_1b_3=a_3b_1 \}$. Prove that $X$ has two irreducible components, and that both of them are of ...
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3answers
251 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
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1answer
42 views

Example of a divisor of a function

I'm studying Fulton's algebraic curves book and on page 97 Fulton defines the divisor of the rational function $z\in k(C)$: I'm looking for an example of a divisor like this one. Thanks
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2answers
168 views

Completion along zero section of an elliptic curve.

I am trying to understand the intuition that I should have about the formal group of an elliptic curve. Say that I have an elliptic curve $E\to \text{Spec} R$ for some ring $R$, with section $0\colon ...
6
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1answer
49 views

Intersection of affine varieties is affine

Let $M,N\subset\mathbb{P}^n$ quasiprojective varieties such that there exist isomorphisms $i\colon M\rightarrow Z(a)\subset \mathbb{A}^m$ and $j\colon N\rightarrow Z(b)\subset \mathbb{A}^m$ for ideals ...
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1answer
41 views

Book which covers these contents of the same level of Fulton's book.

My question is very specific. I'm studying Chapter 8 of Fulton's algebraic curves book and I would like to find another book (or online sources) which covers these contents: Divisors, the Vector ...
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1answer
24 views

under change of coordinates the variety $Z(H_1,..,H_r)$ becomes $Z(x_1,…,x_r)\subset \mathbb{P}^n$.

A set $V\subset \mathbb{P}^n$ is called a linear subvariety of $\mathbb{P}^n$ if it's the zero locus of $r$ homogeneous and linear, i.e $V=Z(H_1,...,H_r )$ where each $H_i$ is a form of degree 1. I ...
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1answer
47 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
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2answers
61 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
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1answer
320 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
3
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1answer
28 views

Exercise 1.13 of II in Hartshorne algebraic geometry.

The problem is following. 1.13 Espace Etale of a Presheaf. Given a presheaf $\mathscr F$ on $X$, we define a topological space $Spe(\mathscr F)$, called the espace etale of $\mathscr F$, as ...
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17 views

Regarding Espace Etale of a presheaf, is $\bar s$ an open map?

I am reading Hartshorne Algebraic Geometry. In the exercise 1.13 of II on page 67, given a presheaf $\mathscr F $ on $X$ , $s \in \mathscr F (U)$, $\bar s : U\to Spe(\mathscr F)$ is defined by $P \to ...
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1answer
120 views

Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
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1answer
61 views

Dimension of moduli space of some stable vector bundles on a cubic 3-fold.

I'm trying to understand the claim that the moduli space of stable rank 2 vector bundles on a (general?) cubic 3-fold, say $X$, with $deg c_2=6$ and $det=O_X(2)$ is of dimension 9. I belive what I ...
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1answer
36 views

A Zariski open subset of a variety has the same dimension as the variety.

I am reading Joe Harris' book Algebraic Geometry: A first course. In the book he says: The Grassmannian $G(k,n)$ contains, as a Zariski open subset, affine space $\mathbb{A}^{k(n-k)}$, and thus ...
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0answers
23 views

Is there a $V$ such that $\operatorname{Pic}(V)\to\operatorname{Cl}(V)$ is not one-to-one?

Does there exist an example of an integral variety $V$ such that the usual map from the Picard group $\operatorname{Pic}(V)\to\operatorname{Cl}(V)$ is not one-to-one?
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1answer
31 views

An injective morphism between varieties that is not an immersion

I believe this is relatively elementary, but I'm struggling to think of an example of a morphism $f: X \rightarrow Y$ between varieties which isn't an immersion in the sense of algebraic geometry. ...
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3 views

Is any F-stable maximal torus contained in some F-stable maximal Borel subgroup?

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
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0answers
43 views

Dimension of affine part equals to the dimension of the variety

Let $\mathbb{P}^n$ be a projective space and let $U_0$ be the affine set with first coordiate nonzero. Take an affine variety $X$ embedded into it. Is it true that the closure of $X$ with the respect ...
4
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1answer
306 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
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20 views

Curves not contained in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface. For instance, a helix is not contained (or non-embeddable) in ...
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29 views

What is the pull back of this line bundle to the effective divisor defining it?

Let $S$ be a surface and $C$ be an effective divisor on $S$. Let $L=\mathcal{O}_X(C)$ be the line bundle corresponding to $C$. Let $i:C\longrightarrow S$ be the inclusion morphism. Then what is ...
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37 views

Punctual Hilbert Scheme of three points

Consider the Hilbert Scheme of zero dimensional, length 3 subschemes supported at the origin in $\mathbb A^2$. Based on this paper (click on Look Inside, the interesting part is on the second page), ...
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2answers
42 views

Is every fiber of a morphism between varieties of pure dimension?

Suppose $f\colon X\to Y$ a morphism of varieties with connected fibers, is it true that all the fibers have pure dimension?
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0answers
42 views

Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

What are the residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$? After finding the maximal ideals of $\mathbb{R}[X,Y]$, which are of the form: $\langle X-a,Y-b \rangle$ with $a,b ...
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0answers
27 views

Inclusion of Tori induces surjection of character groups?

Let $k$ be an algebraic closed field. Let $T, T'$ be algebraic Tori in the classical sense, meaning $T \cong \mathbb{A}_k^n \setminus V(X_1 \cdots X_n)$, $T' \cong \mathbb{A}_k^{n'} \setminus V(X_1 ...
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1answer
47 views

Order of element in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
4
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1answer
22 views

A set is not semialgebraic

A subset $A$ of $\mathbb R^n$ is called semi-algebraic if it can be represented as a finite union of sets of the form \begin{equation*} \{x\in \mathbb R^n\; |\; p_i(x)=0, q_i(x)<0\; \mbox{for all ...
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24 views

Order of elements in algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
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45 views

Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme ...
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22 views

Classification of $\mathbb{G}_m$-torsors?

Is there a nice proof or reference to one of the theorem that any $\mathbb{G}_m$-torsor is isomorphic to $\mathcal{L}\setminus z(S)\to S$? Here I am denoting by $\mathcal{L}\to S$ to be a line bundle ...
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0answers
31 views

Are there invariants of formal group laws other than height?

By a theorem of Lazard, 1-d formal group laws over separably closed fields of char $p$ are classified up to isomorphism by their height. Are there invariants of formal group laws other than height ...
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1answer
52 views

Show that $\mathbb{A}_\mathbb{C}^2 \ncong \mathbb{A}_\mathbb{C}^1 \times_{Spec(\mathbb{Z})} \mathbb{A}_\mathbb{C}^1$

Show that $\mathbb{A}_\mathbb{C}^2 \ncong \mathbb{A}_\mathbb{C}^1 \times_{Spec(\mathbb{Z})} \mathbb{A}_\mathbb{C}^1$ Honestly I don't know where to begin... It's the same as proving that ...
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1answer
30 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
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44 views

Show that $X$ and $Y$ are isomorphic as schemes

Let $X = \{(x,y,z) \in \mathbb{C}^3 : xy=xz=yz=0\}$ be the union of the three coordinate lines in $\mathbb{C}^3$. Let $Y = \{(x,y) \in \mathbb{C}^2 : xy(x-y)=0\}$ be the union of three concurrent ...
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1answer
69 views

Regularity of $k[X,Y,Z]/(Z^2 - f(X)g(Y))$

Let $R = k[X,Y,Z]/(Z^2 - f(X)g(Y))$, for an algebraically closed field $k$ with $\operatorname{char} k\not=2$, and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals $M$ ...
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39 views

Projective curve, local ring, maximal ideal, dimension $k+1$. [closed]

Let $C \subset \mathbb{P}_2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset \mathcal{O}_p(C)$ is its ...
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1answer
67 views

How to show that $\mathfrak{m}/\mathfrak{m}^{2}\rightarrow\mathfrak{m}A_{\mathfrak{m}}/\mathfrak{m}^{2}A_{\mathfrak{m}}$ is an isomorphism?

Let $A$ be a ring and $\frak{m}$ a maximal ideal of $A$. Let $\kappa$ be the field $A/\frak{m}$. How to show that the $\kappa$-linear natural map $$ ...
0
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1answer
44 views

How to prove that $\mathrm{Proj}\left(B/J\right)$ is isomorphic to $\mathrm{Proj}\left(A/J\right)$ if $I\subset J$?

Let $B$ be a graded ring with positive degrees, and let $I$ and $J$ be homogeneous ideals of $B$. We suppose that there exists $N$ such that $I\cap B_{n}=J\cap B_{n}$ for all $n\ge N$. How to show ...