Suppose $H$ is an algebraic group (let's say over $\mathbb{R}$ or $\mathbb{C}$). I'm interested in the space $\mathrm{Sub}_{\text{alg}}(H)$ whose elements are conjugacy classes of algebraic subgroups of $H$. Is it true that $\mathrm{Sub}_{\text{alg}}(H)$ can be realized as a standard Borel...

For a commutative ring $R$ and an $R$-module $M$, let $\operatorname{Gr}(n,M)$ be the Grassmann functor from the category of $R$-algebras to sets. As I understand it, it should be possible to define a subfunctor $U$ when fixing $x_1, \ldots, x_n$ elements of $M$, how does this subfunctor look...

It is known that every finitely generated projective module $M$ over a commutative ring $A$ has locally constant rank, i.e., for each $\mathfrak{p} \in \mathrm{Spec}(A)$, there are non-negative integer $r$ and an open neighborhood $U \subseteq \mathrm{Spec}(A)$ such that, for every $\mathfrak{q}...

I am curious if there is a notion of how to find a Gröbner basis for any ideal $I$ of a finitely generated algebra $R\cong \mathbb{K}[x_1,\dots,x_k]/J$. I know that Gröbner bases are generaly developed as a tool for polynomial rings, but I wonder what fails in this case or in which cases it's...

I'm struggling to understand an argument in the book "Weil Conjectures, Perverse Sheaves, and $l$-adic Fourier Transform" by Kiehl and Weissauer. In Theorem I.6.1, they prove (in specific cases) that the $i$-th cohomology of a pure sheaf of weight $w$ has weight $w+i$. I'm confused by their...

As a soon-to-be senior undergraduate planning to pursue research in Schubert calculus under a supervisor specializing in this field, I have struggled to locate accessible introductory textbooks or lecture notes for this subject, as well as more advanced reference materials to save for my future...

There is a philosophy attributed to Grothendieck that it is better to have a good category (e.g. mapping objects, abelian category, etc) with bad objects than a bad category with nice objects. What are some examples of this? Please also describe some ways these good categories have been helpful.

Let $X$ be a smooth algebraic variety over a field $k$, and let $K/k$ be any field extension. I want to prove that $$ X_K:=X\times_{\operatorname{Spec}k}\operatorname{Spec}K $$ is smooth over $K$. I want to use only the following facts: Jacobian criterion (rational points): If $$...

I fail to understand the highlighted statement in my screenshot below. If $U \subset Y$ is a non empty open subset, then the natural map $g: \mathscr{O}_Y(U) \to k(Y)$ given by $(U,f) \mapsto [U,f]$ is naturally injective. Indeed, if $g((U,f_1)) = g((U,f_2))$, i.e $[U, f_1] = [U,f_2]$, then...

I read little bit about Special Relativity and there was one moment that I can't understand. It was about that there was two irreducible polynomials that have common roots: I was confused because each of these polynomials have different variables. My question is: is it possible that two...

I am self learning Algebraic Geometry from Daniel Perrin's Algebraic Geometry textbook. I have a question on last paragraph of page $41$. Let $D(f)$ be the set of points where the function doesn't vanish and $\Gamma(V)= k[X_1,...,X_n]/I(V)$ where $k$ is a commutative field. Let $r$ denote the...

I am reading up on some algebraic geometry and came across the above definition of sheaves. Some things confuse me. When the author says in 2) of Definition 4.1, "For each inclusion of open sets $V \subset U$..", does he mean that $V$ is an open set in $X$ or in the induced topology on $U$? In...

Problem Statement Let $f: X \to Y$ be a surjective morphism of finite type between affine Noetherian schemes, where $X = \operatorname{Spec} B$ and $Y = \operatorname{Spec} A$. Suppose that for every closed point $y \in Y$ and for every $x \in X_y$ that is closed in $X_y$, the stalk map...

Let $k$ be an infinite field. Definition 1. ( Affine Hilbert function ). Let $R := k[x_1, \dots ,x_n]$ be a polynomial ring which can be viewed as a vector space over $k$. Let $R_{\le s} := k[x_1, \dots, x_n]_{\le s} $ denote the set of polynomials of total degree $\le s$ in $R$. Note that...

This question is from Proposition $6.3$ of the textbook Algebraic geometry by Daniel Perrin( Page 32). Here $k$ is a commutative field.Let $V$ be a projective algebraic set and consider a homogeneous element $f \in \Gamma_h(V)= k[X_0,...,X_n]/I_p(V)$ of degree>0. $I_p(V)$ is the ideal of...

I am self studying Algebraic Geometry from the Clader and Ross Algebraic geometry textbook: Beginnings in Algebraic Geometry. This proof is given on page 275 of the textbook and I am quite confused about it. Please help me. Theorem 9.53 (Projective Nullstellensatz) Assume that $K$ is...

For instance we have the projective space itself is a manifold, and we often talk about zero set of polynomials over the projective space in algebraic geometry. So, how do the non trivial manifold properties /algebraic topology play into the study of algebraic geometry?

Let $R = k[x_1, \dots, x_n]$ be a polynomial ring over a field $k$ equipped with a standard grading, and let $<$ be a monomial order on $R$. Let $I$ be a homogenous ideal of $R$, and let $\operatorname{in}_<(I)$ denote the initial ideal of $I$ with respect to $>$. My question is If $f\notin I$...

Let $Q_N$ denote the cubical cell complex given by the poset of faces of the $N$-cube, and let $$ V_N=\{\pm 1\}^N $$ be its set of $0$-cells. Let $$ A_N:=V_N/\{\pm 1\} $$ be the set of antipodal pairs of $0$-cells. We have $$ |A_N|=2^{N-1} $$ For each antipodal pair $$ \alpha=\{v,-v\}\in A_N $$...

How can I show that every Projective variety in $\mathbb{P}^n$ is compact in the induced Euclidean topology? Should I consider, as customary, an arbitrary open cover in $\mathbb{P}^n$, and perhaps use the projection map $\pi:\mathbb{C}^{n+1}\setminus\{0\}\rightarrow \mathbb{P}^n$ which defines...