# 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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### Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm very intrigued by ...
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### smooth affine algebraic curves and their subschemes

I am reading a lot about curves at the moment and I am a little confused: Let $X= Spec K[X]$ denote a smooth affine algebraic curve. Then, according to some sources, the ring $K[X]$ is a Dedekind ...
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### how to distance circles drawn on another circles

I need to do some calculations in order to do this drawing (sorry for the quick sketch): I need to define a set of variables and do simple calculations as much as possible in order to come up with ...
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### Zariski topology on Affine spaces, Name of Functor

I've been studying the Zariski topology in my free time. So I found this functor between Polynomial Algebras and Affine Spaces. First, we have this $T$ such that for any affine space ...
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### Functor of points definition of a space modeled on a site

I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let $(C,J)$ be a grothendieck site and ...
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### Algebraic families of vector spaces that are pairwise dependent

Let $T \subset Gr(n,2n)$ be an algebraic family of complex $n$-dimensional vector subspaces in $\mathbb{C}^{2n}$, $n > 1$, and denote the vector space corresponding to a point $t$ of $Gr(n,2n)$ by ...
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### Suppose X is a closed subscheme of Y, with Y locally Noetherian. Is there a locally free resolution of $i_* O_X$?

Let $I : X \to Y$ be a closed subscheme of a locally Noetherian scheme. I am secretly trying to show that sheaf exts of $i_* O_X$ to coherent sheaves on Y are coherent (in order to find a dualizing ...
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### Tropicalization of a line in the projective plane P^2

Lets assume that our field $K$ is the Puiseux series. I have been working with tropicalization from the book "Introduction to tropical geometry" link : ...
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### Separable morphism and smooth fibers

Let $f:X \to Y$ be a separable, dominant morphism of finite type between noetherian $k$-schemes for $k$ algebraically closed. Does it mean that For a closed point $x \in X$, $f^{-1}(f(x))$ is smooth ...
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### Example of a dominating map

Unfortunately the book that i am reading (Algebraic curves by Fulton) has no examples, so i am trying to find an example of a dominating map that would be helpful for the understanding of the ...
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### The canonical base point for Weil algebras

Kock defines, after (16.2), the canonical base point of a small object $\operatorname{Spec}_R(W)$ to be $$\mathbf 1\overset{\operatorname{Spec}_R (\pi)}{\longrightarrow}\operatorname{Spec}_RW$$where ...
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### Trying to Compute Regularity and degree

Definition: For a finite subset $X \subset \mathbb P^r$,the Hilbert function $H_X(d)$ is constant for large $d$ and its value is the number of points in X,usually called the degree of $X$. Let ...
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Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? ...
### Problem in understanding the proof of lemma $7.2.5$ in Liu's book
Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
### A basis for forms of degree $d$ (Fulton, 2.35)
I am trying to solve this exercise from Fulton's book: (2.35)(c) Let $L_1, L_2, \dots,$ and $M_1, M_2, \dots$ be sequences of nonzero linear forms in $k[X,Y]$ and assume no $L_i = \lambda M_j$ for ...