# Tagged Questions

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. ...

41 views

### Blow up of an ideal in $\Bbb C^2$

As described in these notes, I am trying to compute the blow up of $\mathbb C^2=\text{ Maxspec }\mathbb C[x,y]$ along the subvariety corresponding to the ideal $\langle\ x^2,y\ \rangle$ but I am ...
51 views

### A version of Bezout's Theorem

I have read the following version of Bezout's Theorem, but I don't get to understand how it implies the classical version. Let $F,G\in K[X_{0},X_{1},X_{2}]$ be non-constant homogeneous polynomials ...
64 views

### Algebraic variety determined by closed points

I am in the process of understanding the importance of the closed points of algebraic varieties (taking the scheme point of view). I ask myself the following question: if varieties $X$ and $Y$ over a ...
32 views

### Closed maps in terms of lifting properties (analogousy to formally étale morphisms)?

In continuation to this MSE question, where closed maps are characterized by "fiber thickenings", I trying to formulate this fiber thickening condition as some lifting property of $f$ against some ...
48 views

### What properties single out $\operatorname{Spec}(\mathbb{k})$-schemes that are quasi-projective varieties over $\mathbb{k}$?

I have a question in algebraic geometry that I would like to ask: Let $\mathbb{k}$ be an algebraically closed field. Is there a property $P$, phrased in the language of schemes, such that ...
24 views

59 views

### Why are the irreducible components $T$-stable?

I'm having trouble with part of a proof (7.1.5) in Springer's Linear Algebraic Groups. Let $r: G \rightarrow \textrm{GL}(V)$ be a rational representation of a linear algebraic group $G$, $B$ a Borel ...
36 views

### Function field, finite extension, isomorphism implies isomorphism?

Let $A$ be a function field in $1$ variable over $\mathbb{C}$, and let $B$ be a finite extension of $A$ of degree $[B : A]$. If $B \cong \mathbb{C}(x)$ over $\mathbb{C}$, then does it necessarily ...
17 views

### Image absolute Frobenius by $G_x \longrightarrow \mathrm{Gal}(\kappa(x)/\kappa(y))$

Let $X$ be a quasi-projective scheme over the finite field $\mathbb{F}_q$ and $G$ a finite group acting on $X$. Let $Y = X/G$ be the quotient scheme and $\pi : X \longrightarrow Y$. Let $x \in X$ be a ...
29 views

### Buchberger algorithm and ideals

I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$. For ...
18 views

### Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description

What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ...
15 views

### Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator \textrm{Ad } : G \rightarrow ...
20 views

### Functoriality of $\text{Proj }A$

This is a remark from Ulrich Gortz's Algebraic Geometry - Functoriality of $\text{Proj }A$ - Let $A=\bigoplus_{d\ge0}A_d$ be a graded ring. We can "thin out" $A$ and "change $A_0$" without ...
86 views

### Understanding the topology of $y^2=(x-1)(x-2)(x-3)(x-4)$

Andreas Gathmann's lecture notes on algebraic geometry start by considering the curve $C_n=\{(x,y): y^2 = (x-1)(x-2)...(x-2n)\} \subset \mathbb{C}^2$. He claims that the topology of this curve is the ...
18 views

### Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
51 views

### Why do we need the infinite field hypothesis in this cohomology calculation?

I've just finished my very first calculation with sheaf cohomology. It's exercise III.2.1(a) in Hartshorne, and it says Let $X = \mathbb{A}_K^1$ be the affine line over an infinite field $K$. Let ...
95 views

### Line bundle trivial on fibers then isomorphic to the pullback of a line bundle

$\require{AMScd}$ I'm currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let $V$ and $T$ be varieties over $k$ with $V$ complete, and let ...
27 views

### Dimension of irreducible component of reduced ring

Let $X= SpecA$ denote the spectrum of a reduced ring $A$. Is there any way to tell the dimension of an irreducible component $Y$ of this variety? Each irreducible component corresponds to a minimal ...
82 views

### Sheaf cohomology with support

Let $X$ be a topological space and $\mathcal F$ is a sheaf of abelian groups on it. Let $Y$ be a closed subspace of $X$. Let $\mathscr{H}^0_Y \mathcal F$ be the subsheaf of $\mathcal F$ with supports ...
67 views

### $\mathbb{P}_{\mathbb{C}}^3$ is not isomorphic to $S^2 \times S^4$

I have been trying to solve this exercise given by my prof. The hint is to show that every $2$-form $w$ on $S^2 \times S^4$ is s.t. $w \wedge w = 0$, while this is not true in case of ...
44 views

### If f/g is symmetric (resp homogeneous), must f and g be as well?

Suppose we have two polynomials $f$, $g$ in $k[X_1, ..., X_n]$ over some field $k$, and they have no factor in common. Suppose that $f/g$ is symmetric. Must than $f$ and $g$ also both be symmetric? ...
40 views

### do formal group laws induce group structures on schemes (as opposed to formal schemes)

Let $R$ be a ring and $f \in R[[x]]$ a commutative formal group law over $R$, meaning $f(f(x, y), z)=f(x, f(y, z))$, $\ f(x, y)=f(y, x)$ and $f(x, y)=x+y + \text{higher order terms}$. Let ...
13 views

### Some clarifications regarding the definition of the Hilbert-Mumford weight

I am reading about the Hilbert-Mumford criterion and I am stuck at something that is stated as "obvious" in every text that I can find. A bit of help would be much appreciated. So, let $X$ be a ...
55 views

### Clifford, $p$-forms and spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here. I'm stuck at the point where it explains how the ...
31 views

### product of divisors on the square of a curve

Let $C$ be a smooth projective curve over a field $k$ and let $D$ be a divisor on $C$. I have seen people considering a zero-cycle on the surface $C \times C$ which they denote by $D \times D$. If ...
26 views

### Why is $T/S$ isomorphic to $k^{\ast}$? (Remark 7.1.4 in Springer Linear Algebraic Groups)

I had a quick question about quotients of varieties. I am still not very good at them. Let $T$ be a torus, $\alpha$ a nontrivial character of $T$, and $S = (\textrm{Ker } \alpha)^0$. Since $T$ is ...
36 views

### How can I define an isomorphism here?

Let $V$ be a variety in $A^n$(affine n-space with algebracally closed field k base) , $I=I(V) \subset k[X_1,X_2,...,X_n], P \in V$, and let $J$ be an ideal of $k[X_1,X_2,...,X_n]$ that contains ...
54 views

### Understanding problem 2.6 in Hartshornes algebraic geometry book .

Can anybody help me in understanding the hint given in the problem $2.6$ in Chapter $1$ of Hartshorne's Algebraic Geometry book ? I cannot see why $A(Y_i)$ can be identified with degree zero ...
32 views

### An algebraic set is called defined over $k$ if its ideal can be generated by polynomials in $k[x]$. [duplicate]

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves. An algebraic set (in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in ...
30 views

### What's the dimension of the space formed by taking a union of projective lines between two spaces?

I've read that the subspace formed by taking the union of all projective lines that join a point of $\mathbb{P}^{l}$ with a point of $\mathbb{P}^{m}$, where $\mathbb{P}^{l}$ and $\mathbb{P}^{m}$ are ...
22 views

### Can Hecke Operators be defined on more general spaces of elliptic curves?

Classically, the Hecke Operators act as endomorphisms of $\omega^k_{\mathcal{M}_{ell}(\mathbb{C})}$, defined by noting that there is a distinguished class of covering maps $\widetilde{E}\to E$ given ...
67 views

### Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point ...
43 views

### How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
40 views

### Fiber of morphism induced by map on stalks

Given a morphism of schemes $f\colon X\to Y$ and a point $x\in X$, the map on the stalks induces a morphism $\operatorname{Spec}\mathcal{O}_{X,x} \to \operatorname{Spec}\mathcal{O}_{Y,f(x)}$. Is it ...
Let $U$ be an open subscheme of $X$ a $\mathbb C$-scheme locally of finite type. I know that $U$ is of dimension $n$ and I have a point $x$ in the closure of $U$. What can be said of the Krull ...
### Exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book
I am trying to solve exercise $2$ from chapter $5$ of Eisenbud's The Geometry of Syzygies book.The problem is as follows: Let $X$ be the union of two disjoint lines in $\mathbb P^3$, or a conic ...