# 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. ...

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### principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Let $I$ a Ideal of $\mathbb{C}[Z,\bar{Z}]$. How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ? It exists some simple criterion to say that an ideal will be principal or not ?
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### Categorical Quotient and group actions

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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### Realization of prequantized Hilbert schemes

Could we define the product of an integral scheme over an algebraic subvariety of positive characteristic if the non-reduced points are not split-solvable over the field? Perhaps a geometric ...
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### Calculating Categorical Quotients [duplicate]

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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### Basis for differentials of a smooth plane curve

Given a smooth plane curve $C$ cut out by a homogeneous polynomial $f(x,y,z)=0$, how to calculate a basis for the space of global differential forms? There is the adjunction formula which shows that ...
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### How can I compute a presentation of the tangent bundle for a smooth manifold defined by a family of polynomials?

Consider a smooth manifold $M$ given by a system of polynomials \begin{align*} f_1 = 0 \\ \cdots \\ f_k = 0 \end{align*} in $n$ variables. This has the algebraic description as the $\mathbb{R}$-...
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### Normalization of schemes which are not reduced

One usually defines normalization for reduced schemes. Is it possible to do it also for non-reduced ones? We know that to any scheme we can associate a reduced one. Is then sufficient to work on this ...
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### Example of non-noetherian ring whose spectrum is noetherian and infinite

A topological space is noetherian if it satisfies the descending chain condition for its closed subsets. Let be $R$ a commutative ring and let $\mathrm{Spec}(R)$ its spectrum with Zariski topology. I ...
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### Proof Check: Prove Relation Between Invariants Is the Only Relation

Consider the finite matrix group $C_{4} \subset$ GL$(2,\mathbb{C})$ generated by $$A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \in \text{GL}(2,\mathbb{C}).$$ (a)Prove that $C_{4}$ is ...
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### What properties are preserved by direct limits? [on hold]

We know that direct limit of a directed family of flat $R$-modules is also flat ($R$ is a commutative ring with $1$ and all modules are unital). I am looking for other properties of modules which ...
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### How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
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### On the definition of a reductive group.

Wikipedia defines a reductive group $G$ as an algebraic group with trivial unipotent radical. The radical is the connected component of identity in the maximal normal solvable subgroup of $G$. The ...
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### Complex conics as a Riemann surface

Consider the complex curve defined by $\{(x,y) \in \hat{\mathbb{C}}^2 | ax^2 + bxy + cy^2 + dx + ey + f = 0 \}$ for some complex numbers $a,b,c,d,e,f$ (here $\hat{\mathbb{C}}$ is the Riemann sphere). ...
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### Is the product of algebraic groups the same as the fibre product?

Assuming we have two algebraic groups $G_1$ and $G_2$ over $k$. Then the direct product $G_1 \times G_2$ with the direct product group structure is an algebraic group. Is this the same as the fibre ...
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### Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
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### How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
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### Duality between cut ideals and cycle ideals?

There exist a general duality between vertex-cuts and cycles and also Duality Principle on Digraphs. I am trying to find a duality prienciple expressed in terms of ideals so Does there exist a ...
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### Moduli Space of Hyperelliptic Curves as Fibration?

Basically, I had a thought about a way to think of the moduli space of hyperelliptic curves. I'm sure it's wrong most likely, but I was hoping someone could maybe point out the flaw in my reasoning. ...
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### Example of an monomial ideal that is weakly reverse lexicographic but not reverse lexicographic

We are looking at a paper titled "Generic Ideals and Moreno-Socias Conjecture" by Edith Aguirre, et al. In the paper they state that an ideal which is reverse lexicographic is also weakly ...
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### Example $3.3.1$ in Hartshorne

Let $k$ be an algebraically closed field, and let $$X = \operatorname{Spec} k[x,y,t]/(ty-x^2)$$ $$Y = \operatorname{Spec} k[t]$$ Hartshorne comments that both schemes $X$ and $Y$ are of finite type ...
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### Ext group of bundles on moduli space of curves

Let $\mathcal{M}_{g}$ be the moduli space of curves of genus $g$. Let's suppose $g \geq 2$. Let $T$ be the tangent bundle of $\mathcal{M}_{g}$. Is the Ext group $\text{Ext}^1(\bigwedge^2T, T)$ trivial?...
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### Planar ternary ring point operations

I have the following topic in my exam questions' list: Prove that point operations in a planar ternary ring satisfy field axioms. I know Proposition 1 from this paper but this only says something ...
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### Endomorphism ring of an abelian variety and its reduction mod $\mathfrak{p}$

Let $A$ be an abelian variety defined over a number field $K$. Let $\mathfrak{p}$ be a prime of $K$ for which $A$ has good reduction and let $k=\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}$. Let \$\...