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
15 views

Why is conjugation by a diagonal matrix a semisimple automorphism of $\textrm{GL}_n$?

Let $$s = \begin{pmatrix} \lambda_1 & & & 0\\ & \lambda_2 & \\ & & \ddots \\ 0& & & \lambda_n \end{pmatrix}$$ be a diagonal invertible matrix. Let $G = ...
6
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1answer
158 views

Is there a (not so) generalized version of Hilbert's Theorem 90?

I'm sorry if my following question doesn't make any sense. We know that if $L/k$ is a finite Galois extension then $H^{1}(\mathrm{Gal}(L/k),L^{*})=0$ (Hilbert's theorem 90). However I would like to ...
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0answers
46 views

How to show $\text{Sym}^n(\mathbb{P}^1)=\mathbb{P}^n$

From the question Theon Alexander (http://math.stackexchange.com/users/165460/theon-alexander), Reference-Request: Symmetric Product Schemes, URL (version: 2014-08-10): ...
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17 views

Symmetric Product of a Projective scheme

Following the question, Theon Alexander (http://math.stackexchange.com/users/165460/theon-alexander), Reference-Request: Symmetric Product Schemes, URL (version: 2014-08-10): ...
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0answers
12 views

A function vanishing on the subgroup generated by relations defines a linear function.

I am reading Basic Algebraic Geometry 1 by Shafarevich (3rd edition) and I couldn't understand the following portion on pg 222: Namely, in Section 5.2 we defined the module of differentials ...
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1answer
26 views

Solving nonlinear system algebraically

I have the system of equations: $$2x(1+\lambda)=0$$$$2y(1+\lambda)=0$$$$2z(1-\lambda)=0$$$$x^2+y^2-(z^2+1)=0$$ It's easy to plug in a few values and see that the solution is $x^2+y^2=1$, $z=0$, and ...
2
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1answer
28 views

Show that this morphism of varieties is not separable

Let $k$ be an algebraically closed field of characteristic $2$, $G = \textrm{SL}_2(k)$, and $z = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$. Let $\sigma: G \rightarrow G$ be the ...
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1answer
52 views

Show $Z(yf-1)$ is irreducible.

Question: $k$ is an algebraically closed field. Let $f \in k[x_1, \ldots, x_n]$ be an irreducible polynomial. Show that $Z(yf-1)\subseteq \textbf{A}^{n+1}$, with coordinates $x_1, \ldots, x_n, y$, ...
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0answers
24 views

Dimension of the fibre product of varieties

Let $\mu : X \to Y$ be a morphism of algebraic varieties. Suppose that both $X$ and $Y$ are of pure dimension $x$ and $y$, respectively. Under what assumptions can we conclude that the fibre product ...
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22 views

Universal family of the Hilbert scheme of one point.

If $X$ is projective over a base scheme $S$, then its Hilbert scheme of one point is just itself. i.e. \begin{equation} \text{Hilb}^1(X)=X \end{equation} What about the universal family? i.e. there ...
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30 views

Hilbert scheme of $n$ points on a smooth curve

If $C$ is a smooth curve over a field $k$, then from lots of references, e.g. Janos Kollar, Rational Curves on Algebraic Varieties, exercise 1.4.1, that the Hilbert scheme of $n$ points is ...
2
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1answer
55 views

Image of divisors under étale cover

Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties. If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism ...
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34 views

Dual of lisse complexes

Let $X$ be scheme of finite type over a field of characteristic $p$. Let $L$ be a lisse bounded complex of étale $\mathbf{Z}/l$-sheaves (with $l$ a prime different from $p$). I remind you that this ...
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1answer
32 views

Visualizing sections of nontrivial vector bundles

My question is simply: how does one think about sections of nontrivial vector bundles on a smooth manifold, for example? The canonical example I think of is a vector field, i.e. a section of the ...
2
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1answer
33 views

Morphism whose fibers are finite and reduced is unramified

Definition: Let $f: X \to Y$ be a morphism of finite type of locally Noetherian schemes, $x \in X, y = f(x) \in Y$. Say that $f$ is unramified at $x$ if the map on stalks satisfies $m_y ...
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0answers
17 views

Image of map of étale fundamental groups

Let $X$ be a connected (locally Noetherian) scheme and $\phi: Y \rightarrow X$ be a finite etale Galois cover of degree $d$, i.e. $Y$ is connected and there are $d$ $X$-invariant automorphisms of $Y$. ...
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30 views

Existence of a certain nodal quartic curve

I am reading this (https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/Univcodim2invent.pdf) paper of Voisin, but I am having some trouble with the proof of Sublemma 2.8 (it might be something very ...
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0answers
32 views

When does a partially defined map from a singular curve to projective space extend?

This is just something train of thought I became curious about after reading a cryptic suggestion that resolution of singularities lets us transfer our "understanding" of smooth curves to singular ...
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1answer
48 views

Can every polynomial generate an ideal?

Suppose an arbitrary polynomial $f$ in a polynomial ring $R$. Is $\langle f\rangle$ always an ideal? Helper parts Consider a finite polynomial ring. Let $R=R[x_1,\ldots,x_n]$. Is the answer ...
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30 views

How is “Binomial” defined in Algebraic Geometry?

I am learning ideal arithmetics and I was flabbergasted that $\langle x\rangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read paper ...
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28 views

Binomial ideals that are not toric i.e. binomial and not prime?

Let $R$ be a ring. Toric ideal $I$ is binomial (generated by a binomial) and prime (the quetient ring $R/I$ is integral domain). Paper and corollary 1.3 is to determine whether an ideal is binomial. ...
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43 views

How to construct the map relating topological and algebraic $K$ - theory ?.

Let $X$ be a complex projective variety, and let $ X^{an} $ denote the topological space of complex points of $ X $ equipped with the analytic topology. Then, any algebraic vector bundle $ E \to X $ ...
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67 views

Embedding of Kähler manifolds into $\Bbb C^n$

Consider $\Bbb C^n$ with its standard hermitian product. This space produces many example of Kähler manifolds simply by taking a smooth affine variety $X\subseteq\Bbb C^n$ with the induced metric. Now ...
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67 views

Is $(\mathbb R^d/\mathbb Z^d,+)$, $d>2$, isomorphic to some group of an algebraic surface?

It is a well-known fact that for points of a cubic curve over $\mathbb{RP}^2$ we can define a group $(G_{\mathbb{RP}^2},+)$ using Cayley–Bacharach theorem. See Wiki: The group law. Another fact ...
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1answer
62 views

The Zariski closure of a constructible set is the same as the standard closure?

Question: Let $X$ be an affine variety over $\Bbb C$, and let $Y\subseteq X$ be a constructible set (i.e. $Y$ is a finite union of locally closed sets). Is it true that the Zariski closure of $Y$ ...
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0answers
67 views

Canonical scheme structure on the singular locus of a variety

Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations $$X:(F_1=\cdots = F_m=0)\subset \mathbb{A}^n.$$ Then $X$ is singular at $p\in ...
2
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1answer
53 views

Has toric ideal something to do with torus?

I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of "toric ideal". Is there a geometric ...
2
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0answers
69 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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70 views

Is $X$ an algebraic subset? Analytic subset?

Suppose that $X$ is a subset of $\mathbb{C}^n$, and that every (complex) hyperplane section of $X$ is an algebraic subset (respectively analytic subset) of complex dimension at least one (or empty). ...
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31 views

Fan associated to a rectangle

In this question, there is a claim that the fan corresponding to the square in $\mathbb{R}^2$ consists of the 4 quadrants - but I think this is wrong or I am confused. The faces of the square are ...
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0answers
25 views

Stratifications and constructible sheaves

Let $S$ be a reduced scheme of finite type over a perfect (even finite or algebraically closed if you want) field and let $F$ a constructible $\overline{\mathbf{Q}}_l$-sheaf over $S$. Why can we ...
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1answer
31 views

Order and residue of 1-form $x^{-1}dx$

Consider the rational 1-form $x^{−1}dx$ on $\mathbb{P}^1$. I am asked to compute its order and residue at all $P \in \mathbb{P}^1$. Could somebody help me with this? I do not really how to start ...
2
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1answer
31 views

A-basis of $\Omega_A^1$

In the lecture notes of a course in Algebraic Geometry that I am following they are calculating a basis of $\Omega_A^1$ for $A=k[x_1,...,x_n]/(f_1,...,f_r)$. We get 􏰎 \begin{align*}\Omega^1_A = ...
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1answer
51 views

Why is this vanishing set nowhere dense? [closed]

Let $A$ be a commutative ring and $f\in A$ be a nonzerodivisor. Why is $\mathrm{V}(f)$ nowhere dense in $\mathrm{Spec(A)}$?
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40 views

dimension and intersection

Let $X$ be a scheme (say of finite type over a field), $i: Y \hookrightarrow X$ a closed immersion and $j : U \hookrightarrow X$ be the open complement. Let $F$ be an étale sheaf over $X$ ...
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0answers
51 views

Schubert calculus: lines that intersect 4 given curves

I am reading these notes on Schubert calculus: http://www.jasoncantarella.com/downloads/laksov.pdf and I don't understand the example about lines intersecting given curves $C_{1},C_{2},C_{3},C_{4}$ on ...
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0answers
25 views

Quotient of a scheme under infinite group action

Let $X=\mathbb{A}^2\setminus\{(x,y)\}$ be the affine scheme minus the origin (say over a field $k$). Consider the action of the group $G=k^*$ given by $(x,y)\mapsto (\alpha x,\alpha^{-1}y)$ for any ...
0
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1answer
34 views

Locally finitely generated sheaf

Here is the image from the book So I don't quite understand the proof of lemma 3.9. Namely, I don't see why there exists $H_{jk}$ such that the formula is true on $U'$. I was wondering if someone ...
3
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1answer
155 views

Alternative description of the sheafification

For me the sheafification of a given presheaf is this: Proposition-Definition: Given a presheaf $\mathscr{F}$, there is a sheaf $\mathscr{F}^+$ and a morphism $\theta \colon \mathscr{F} \to ...
6
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1answer
51 views

Vakil's Foundations of Algebraic Geometry, Exercise 7.3.F

$\DeclareMathOperator{\Spec}{Spec}$ I'm having trouble with the exercise in the title, which states "Suppose $Z$ is a closed subset of an affine scheme $\Spec A $ locally cut out by one equation. ...
8
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0answers
168 views

Is an étale morphism of algebraic stacks locally quasi-finite?

An étale morphism of schemes is unramified, and an unramified morphism is locally quasi-finite. Does the same hold for étale morphisms of algebraic stacks? Let us recall the definitions, following ...
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1answer
34 views

Is there an example of an algebraic surface with a (-2)-curve and…

Recently I am curious about the minimal model program on varieties of log canonical Calabi-Yau type over complex numbers. That is, a log canonical pair $(X,B)$ such that $K_X+B \sim_Q 0$. Given such a ...
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43 views

Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
2
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1answer
31 views

The set of commutative matrices is an irreducible algebraic variety

Let $A, B$ matrices $n \times n$. Let $X = \left\{(A, B) \in \mathbb{A}^{2n^2} \mid AB = BA \right\}$. Prove that $X$ is algebraic and irreducible variety.
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41 views

How to compute explicitly the covering map in the modularity theorem?

The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $\pi:X_0(N) \to E$ for every $E$, an elliptic curve defined over ...
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0answers
28 views

projective nullstellensatz proof

I have to give a talk about projective varieties, including the projective nullstellensatz. As I'm not really into algebra or algebraic geometry, I've got some problems with the proof. Projectiv ...
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1answer
37 views

Value of polynomials in quotient ring

Let $K$ be a field. If we have a polynomial ring, $K[X_1,...,X_n]$, and an ideal $I$, we can form the quotient ring, $$K[X_1,...,X_n]/I.$$ For a given ideal, $I$, if we take an element of this ...
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1answer
41 views

Is $K^{n}$ Zariski Hausdorff when $K$ is a finite field?

Assume that $K$ is a finite field. Is it true to say that $K^{n}$ is a Hausdorff topological space with Zariski topology?
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34 views

Base-point free linear sheaf and extending the base field

This is a question from Ravi Vakil's notes I've been stuck on, namely 18.2.I. Let $X$ be a scheme over a field $k$, and let $K/k$ be any field extension. Let $\mathcal{L}$ be an invertible sheaf on ...
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1answer
52 views

Geometric xample of a one object cover which does not satisfy the sheaf condition?

For topological spaces, a one object cover automatically satisfies the sheaf axiom because open inclusions are injective, which is equivalent to the projections of the kernel pair (= self ...