Let $$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 $$ be a short exact sequence of vector bundles. It is well known that for exterior powers there exists a natural filtration $$ 0=F_{r+1}\subset F_r\subset \cdots \subset F_0=\bigwedge^r B $$ such that $$...

Consider a non-cuspidal point $x\in X_0(p)(K)$ representing an elliptic curve $E/K$ over a quadratic field $K$, and suppose that $E$ has potentially multiplicative reduction over a prime $\mathfrak{q}$ of $K$ over the rational prime $q$. Let $0,\infty$ be the cusps of $X_0(p)$. Using the...

This is about the umpteenth discrepancy of definitions in algebraic geometry. It's about recipes to define properties of morphisms of algebraic spaces out of properties of morphisms of schemes. When the morphism of algebraic spaces in question is representable the recipe is clear [SP, 025V;...

We say that a topological space $X$ is irreducible if it can not be written as the union of two proper closed subsets. My question is: Do we consider this definition when the topological space is defined in terms of open subsets or closed subsets? When working with the zariski topology, the...

I am reading Arithmetic Moduli of Elliptic Curves by Katz and Mazur, and I have a question about the beginning of Chapter 2, §2. Let $f:E\to S$ be an elliptic curve. Since the sheaf of relative diffrentials $\Omega^1_{E/S}$ is an invertible sheaf on $E$, one defines...

Let $A$ be a commutative ring and $f, g \in A[S, T]$ be two homogeneous polynomials in two variables of homogeneous degrees $d$, resp. $e$. Their resultant $\newcommand{\Res}{\mathrm{Res}}\Res(f, g)$ is defined to be the determinant of the linear map of free modules of rank $d + e$ $$ (f, g)...

So recently the Jacobian conjecture has been disproven. The counterexample had a determinant of $-2$. See for instance : https://www.newscientist.com/article/2580374-ais-solution-to-87-year-old-riddle-takes-mathematicians-by-surprise/ or Wikipedia. Now I wonder if this polynomial can lead to an...

I notice that the recently-publicized counterexample(s) to the Jacobian Conjecture are not surjective. Is that necessarily the case? That is, (Q) If $F:\mathbb C^n \to\mathbb C^n$ is algebraic and everywhere locally injective, and also surjective, must it be injective? Maybe the relevant setting...

Let $X,Y$ be two algebraic surfaces (=smooth, proper $2$-dim schemes over fixed base field $k$) and let $E \subset X$ exceptional curve, ie $E \cong \Bbb P^1$ with self intersection $E^2=-1$. By Castelnuovo's contraction theorem $E$ can be contracted to a smooth point of a smooth surface leaving...

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...