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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Help with Math software (macaulay 2)

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...
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23 views

De Rham-Weil theorem

I am having trouble understanding a couple of points with regard to the De Rham-Weil theorem and was hoping that someone might be able to shed some light. Let $X$ be a smooth manifold and ...
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1answer
26 views

Algebraic Curves and Second Order Differential Equations

I am curious if there are any examples of functions that are solutions to second order differential equations, that also parametrize an algebraic curve. I am aware that the Weierstrass $\wp$ - ...
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1answer
27 views

Can two cuboids with different side lengths have the same volume and perimeter?

We know that two rectangles with different side lengths cannot have the same area provided their perimeter is the same. But can two cuboids with different side lengths have the same volume and ...
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1answer
31 views

Is the closure of a set $A$ in the Zariski topology $V(A)$?

Let $R$ be a commutative unitary ring and $A\subseteq \operatorname{Spec}(R)$ a subset. If $A=\{p\}$, then the closure $\bar A$ of $A$ equals $V(p)=\{x\in \operatorname{Spec}(R) \mid p\subseteq x\}$. ...
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31 views

Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
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30 views

Confusion over common criterion for a line bundle on a scheme to be trivial?

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to ...
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1answer
25 views

find equally spaced points on parabola

I'm trying to find equally spaced points on a parabola simply defined by $$y = \frac{x^2}{2 p}$$ Someone told me there is an easy way to split the parabola but he didn't tell me how and I cannot find ...
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2answers
52 views

Prerequisite of Algebraic Geometry

Algebraic geometry, as far as I know, is a very important branch of mathematics, which is also very difficult. I am going to take a try to taste that. Before really going into the field, I have two ...
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1answer
52 views

What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course ...
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54 views

are connnected components of this scheme irreducible?

So I have a normal surface (ie, all components are dimension 2) $X$ which is smooth and affine over a Dedekind domain $R$ (so $X$ is an affine scheme). Suppose $R'$ is integral (possibly not dedekind) ...
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57 views

What are some applications of anabelian geometry outside of mathematics? [on hold]

I want to learn about some applications of anabelian geometry outside of mathematics. Practical things from any area are welcomed.
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24 views

Inverse limit of irreducible spaces

Let $(X_{i})_{i \in \mathbb{N}}$ be an inverse system of topological spaces. Assume that each of the $X_{i}$ is irreducible. Then is it true that $\projlim X_{i}$ is also irreducible? I read in a ...
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1answer
24 views

Question about characteristic polynomial of the Frobenius endomorphism on elliptic curves.

I have another possibly trivial question about elliptic curves. A lot of papers I've seen state that the characteristic polynomial of the Frobenius endomorphism of an elliptic curve over a finite ...
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1answer
40 views

Group theory and Complex Analysis

Actually this time I am having interest in group Theory and complex analysis. So I want to study the topics which relate both. So please suggest me which topic or book should I read to explore the ...
2
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1answer
38 views

A question about endomorphism rings of elliptic curves

This is probably a very trivial question, but I haven't been able to find a rigorous explanation anywhere so far or at least haven't understood it. Assume we have an elliptic curve $E$ over ...
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1answer
27 views

Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings?

A brief question, if $\pi\colon E\to X$ is a vector bundle over a scheme $X$, is it automatic that any section $s\colon X\to E$ always a closed embedding?
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0answers
18 views

are extensions of flat connections flat?

Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a ...
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2answers
59 views

Duality and Serre's criterion

Let $X$ be a projective scheme, $\mathcal{F}$ a coherent sheaf on $X$ which is $S_2$. Then under what additional conditions is its dual, $\mathcal{H}om_X(\mathcal{F},\mathcal{O}_X)$ also $S_2$?
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1answer
38 views

what is the precise definition of a morphism defined over $k$?

Let $k$ be a field and $X$ an algebraic variety over $k$. I have often seen people write $Aut(X/k)$ for the automorphisms "defined over k". What is the exact definition? My guess is that $X$ comes ...
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1answer
32 views

Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings?

Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings? Is it of finite type? Is the map $\operatorname{Spec} \prod_{i=1}^\infty \mathbb{C} ...
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0answers
18 views

Double Coset Space

In the theory of Shimura varieties you construct the variety as a double coset space $G(\mathbb{Q}) \backslash X \times G(\mathbb{A}_f) / K$ . I do not understand this as a variety. I thought you ...
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2answers
38 views

Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
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2answers
48 views

What's wrong with my example?

I have been asked to show that if $V\subset k^n$ is an affine algebraic variety over an algebraically closed field $k$, and $dom(f) = V$ for some $f\in k(V)$ then $f$ lies in $k[V]$. Here, $k(V)$ is ...
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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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71 views

About the actual state of research in the field of Hodge conjecture. [on hold]

Good evening everyone, Can someone inform me about the current state of research in the field of the Hodge conjecture ? Where are we currently stops in mathematical research in this area? Thank you ...
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2answers
54 views

Sheafification, stalks and quotient

I gave a problem that I can't finish by myself. Any help would be appreciated. Consider a sheaf $\mathcal{F}$ of abelian groups on a topological space. I would like to show that given two sheaves ...
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0answers
31 views

dimension of the span of all partial derivatives of a given polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...
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1answer
36 views

Does a point on an affine curve have to be non-singular for the local ring to be integrally closed?

I am not sure how the non-singularity requirement would come into the following argument: Suppose $\mathcal{C}_f$ is an irreducible affine curve with local ring $\mathcal{O}_P$ at a general point $P$ ...
2
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0answers
38 views

Calculating eigenvalues of the induced action on $H^0(2 K_C)$

Given a (smooth) curve $C$ and an automorphism $\phi$ of $C$. In the first part of their paper On the Kodaira dimension of the moduli space of curves Harris and Mumford calculate the eigenvalues of ...
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43 views

A confusion on the definition of morphism between varieties

I'm currently read the book The Arithmetic of Elliptic Curves by Silverman, he defines the morphism between variety $V_1$ and $V_2$ to be the rational map: $$\phi=[\phi_0,\phi_1,\cdots,\phi_n],\text{ ...
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1answer
30 views

locally ringed space $(X,\mathcal{O}_X)$ isomorphic as Ringed Spaces to $Spec(A)$ but not isomorphic as Locally Ringed Spaces.

I'm starting studying Hartshorne Chapter II and is the first time that I'm studying Schemes. I'm looking for some intuition viewing some examples. I'm looking for an example of a locally ringed ...
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1answer
57 views

Describe the closure in the Zariski topology

I have a little bit to fight with the Zariski - topology. In $\mathbb{C}^2$, i have to describe by a finite number of polynomial equations the Zariski closure of: $C:= \{(n,n^2) \in \mathbb{C}^2 : ...
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1answer
23 views

Vanishing set of irreducible polynomials

Question: Find irreducible $f,g \in \mathbb{R}[x,y]$ such that $V(f) = V(g) \neq 0$ with the added requirement $f \neq \lambda g$ for $\lambda \in \mathbb{R} - \{0\}$. Attempt: I think $f(x,y) = x^2 ...
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1answer
21 views

Does the nilpotent extension of a $1$-dimensional algebra always give a projective module?

Let $A$ be a $1$-dimensional reduced Noetherian algebra over an algebraic closed field $k$ with characteristic zero. Let $(B,N)$ be a nilpotent extension of $A$, i.e. $B$ is a Noetherian $k$-algebra, ...
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0answers
31 views

Is this evaluation map injective?

Let $S$ be a surface, and $E$ be a vector bundle of rank $r+1$, with $h^2(S,E)=h^1(S,E)=0$. Consider $\Lambda\in Gr(r+1,H^0(S,E))$, and the evaluation map $e:\Lambda\otimes\mathcal{O}_S\longrightarrow ...
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0answers
23 views

Automorphisms of del Pezzo surface

Let S - del Pezzo surface of degree 1 (of degree 2), $k = \mathbb{C}$. I don't know how to proof that groups of automorphisms Aut(S) is finite group.
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1answer
34 views

Space of quintics in $\mathbb{P}^3$ that contain intersection of two quadric surfaces

Let the curve $C$ in $\mathbb{P}^3$ is an intersection of two quadric surfaces in $\mathbb{P}^3=\text{Proj}(S)$, $S=k[x_0,x_1,x_2,x_3]$. We have the following resolution of the ideal sheaf ...
2
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0answers
26 views

Regular functions on algebraic set vs localization

Let $V$ be an algebraic set of $\mathbb A^n$ (i.e. a Zariski-closed subset of $k^n$, for some field $k$). As usual a function $\phi : V \to k$ is regular at $p \in V$ if in some neighborhood of $V$ it ...
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1answer
46 views

How does one interpret an element of k[V]?

If $k$ is an algebraically closed field and $V$ is an affine variety, what does it mean for some $f$ to lie in $k[V]$? Can someone give an example of such an $f$? Thanks, Kartik
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0answers
30 views

Complex singularity exponent

I am studying about complex singularity exponents of holomorphic functions. I need some help to clarify a few things: First, what a complex singularity exponent is, for the holomorphic function ...
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0answers
37 views

Two questions about the proof of Proposition V.2.2 in Hartshorne's Algebraic Geometry

This proposition on page 370 is to prove any ruled surface over a nonsingular curve $C$ is $\bf{P}(\mathscr E)$, the projective space bundle of a locally free sheaf $\mathscr E$ of rank $2$ on $C$ and ...
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2answers
42 views

Form of maximal ideals in an algebraicaly closed polynomial ring

I have been trying to prove the following bijection which is a consequence of the nullstellansatz $$\{\text{maximal ideals of }\mathbb{C}[x_1,\dots,x_n] \} \leftrightarrow \{\text{points in ...
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31 views

Calculating Hodge numbers by means of locally free resolutions

In this paper the author considers a smooth $3$-fold $X$ in $\Bbb{CP}^6$ with the following locally free resolutions of its structure sheaf and squared ideal sheaf: $$0\to \mathcal O_\Bbb {P^6}(-7) ...
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0answers
40 views

Family of quartic surfaces in $\mathbb{P}^3$ that contain a fixed line or conic

Let $V$ be a complex vector space of dimension 4 and $\mathbb{P}^3=\mathbb{P}(V)$. The space of quartic surfaces in $\mathbb{P}^3$ is $\mathbb{P}(\text{Sym}^4(V^*))$ and the dimension of this space is ...
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1answer
36 views

Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
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1answer
23 views

Definition of a fan of a polytope

In Fulton's book Introduction to Toric varieties (page 25), he says that: A rational convex polytope $K$ in $N_{\mathbb{R}}$ determines a fan $\Delta$ whose cones are the cones over proper faces ...
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77 views
+100

Proving exactness of the conormal sequence

Problem: Let $\phi \colon A \to B$ be a surjective homomorphism of $R$-algebras with kernel $I$. I want to show that the conormal sequence $$ I/I{}^2 \longrightarrow B \otimes_A \Omega_{A/R} ...
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1answer
27 views

Unicity of a projective transformation determined by 5 points in $CP^3$?

Consider an ordered set of five points $\{p_1, p_2, \dots, p_5\}$ in linear general position in $\mathbb{CP}^3$ and another ordered set of five points $\{q_1, q_2, \dots, q_5\}$, also in linear ...
3
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1answer
31 views

$(A_f)_{g/f^{n_0}}\cong A_{fg}$ (localization with the powers of an element)

I'm working in a problem from Hartshorne Algebraic Geometry. But I need a result from Commutative Algebra. Given a commutative ring $B$ with $1$. For each $b \in B$ define the ring $B_b$ as the ...