I am currently studying algebraic geometry and trying to understand projective spaces. let $S = \mathbb{Z}[x_0, \dots, x_n]$ so that $\mathbb{P}^n_{\mathbb{Z}} = \operatorname{Proj}(S)$. I read that if we remove the standard open affine subset $D_+(x_i) = \{ \mathfrak{p} \in...

I'm studying algebraic geometry and have a question on Kahler differentials. Let $k$ be a commutative ring with unit and $A$ a $k$-algebra. I already have the intuition if we picture $A$ as some ring of functions on $\operatorname{Spec}{A}$, the the module of relative Kahler differentials...

I'm self studying commutative algebra. Here is the question: Edit(Background and Definition): Let $k$ be a field, we define an affine variety $V\subseteq k^n$ as the set of common zeroes of some polynomials $f_1,\dots, f_m\in k[x_1,\dots, x_n]$, define its coordinate ring as $k[x_1,\dots,...

Given a real polynomial $f(x)=a_0x^d+\cdots+a_1 x$, I want to give a sharp estimate for $\sum_{n\le X} e(f(n))$. If $f(x)=ax$ is a linear function, we have $$\sum_{n\le X} e(\alpha n)\ll \min (X,\|\alpha\|^{-1});$$ If $f(x)\in\mathbb{Z}[x]$ where $p$ is a prime, using Weil's bound for...

https://en.wikipedia.org/wiki/Rational_mapping The usual example is that $ \mathbb {P} _{k}^{2} $ is birational to the variety $ X $ contained in $ \mathbb {P} _{k}^{3} $ consisting of the set of projective points $ [w:x:y:z] $ such that $ xy-wz=0 $, but not isomorphic. Indeed, any two lines in...

This question follows from Lemma 6.1.1 from CH 6.1 of Field Arithmetic by Fried and Jarden. It is the subsection on Decomposition groups.' The following paragraph sets up the notation used. In the construction of Decomposition groups the chapter starts by defining $R$ to be an integrally closed...

Background The triangular grid places integer $T(r-1)+c$ at row $r$, column $c$, where $T(n)=n(n+1)/2$. Transposing this grid, reading along SE diagonals of the triangular grid as columns, yields a new array whose column $d$ has values $$f_d(n) = T(n+d-2)+n = \frac{n^2+(2d-1)n+(d-1)(d-2)/2 + ......

About prime factorization of consecutive integers, we all can notice prime factors vary apparently without any logic. Some numbers like $82 = 2 \times 41$ have highly unequal factors (high variance among the factors), while others like $80 = 2^4 \times 5$ or $2310 = 2 \times 3 \times 5 \times 7...

I am preparing a presentation and would appreciate feedback on the following explanation connecting the multiplicative group with Lubin–Tate theory. Let $K$ be a field and consider elements $x,y\in K^\times$ near the identity $1$. Write $$ x=1+X,\qquad y=1+Y. $$ Then the group law on $K^\times$...

Given a multiplicative function $f$ with divisor bound $|f|\le \tau_k$, where $k$ is a nonnegative real number. We consider the Dirichlet series $$ F(s)=\sum_{n=1}^\infty \dfrac{f(n)}{n^s}. $$ Since $$ \sum_{n=1}^\infty \dfrac{\tau_k(s)}{n^s}=\zeta(s)^k, $$ $F(s)$ absolutely converges on the...

Consider affine families $$ Q(u)=2^t3^{16}u+B, $$ with $t\ge 3$ and $2^t\mid 3B-1$. Set $$ C_0=\frac{3B-1}{2^t}. $$ Then $$ 3Q(u)-1 =2^t3^{17}u+(3B-1) =2^t(3^{17}u+C_0). $$ Suppose we restrict to a subfamily where, after the fixed factor $2^t$, another $2^\lambda$ divides the remaining factor:...

$\require{AMScd}$I am in the process of understanding how to work with (quotient) stacks. In my case, it is usually helpful to get my hands dirty so I like to come up with examples where maybe using stacks can make an argument more transparent. I recalled an exercise that I did when preparing...

Consider functions from $R^n$ to a subset of $R^n$. So $f(x_1,x_2,...,x_n) = (y_1,y_2,...,y_n)$. where $x_i,y_i$ are all real. More specific consider $$f(x_1,x_2,...,x_n) = (Q_1(x_1,x_2,...,x_n),Q_2(x_1,x_2,...,x_n),...,Q_n(x_1,x_2,...,x_n))$$ where the $Q_i$ are all Quadratic forms. Even more...

Numbers that are not yet mapped to are marked prime and given their own "trees", but really they are distance $\infty$ from the other primes. Traditionally, in a connected graph, BFS forms one tree, but really this is a collection of overlapping trees. What we have on the number line is a...

$$3(a^3+b^3+c^3+a+b+c)+5(a^2+b^2+c^2)+2(a^2b+b^2c+ac^2)+4(a^2c+bc^2+ab^2+ab+bc+ac)=0$$ A table of primitives known to me $(a, b, c)$, $a \in{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}^+$ $$ \boxed{\begin{array} {|r|r|r|r|}\hline №(a,b,c)& a_n & b_n & c_n \\ \hline S_1 & 0 & 0 & 0 \\ \hline S_2 & 0...

Let $$ \mathcal S=(\mathcal I,\Gamma,\Pi) $$ be a seam marked seed built from four compact oriented $2$-dimensional complex orbifold sheets. Each sheet is assumed to be a football type orbifold: its coarse underlying Riemann surface is $$ |\mathcal O_i|\cong \mathbb P^1 $$ and it has two...

Let $A\in GL(r,\mathbb{Z})$ be a positive definite matrix, $Q(x)=\dfrac{1}{2} x^t A x$ be the quadratic form associate to $A$, $Q^*(x)=\dfrac{1}{2} x^t A^{-1} x$ be the adjoint form of $Q(x)$ . Given $(c,d)=1,m\in\mathbb{Z}^r,$ we define $$G_m\left(\dfrac{d}{c}\right)=\sum_{h\,\text{mod}\,c}...

I apply the iterated forward difference operator to the sequence of primes; from $$ (p_n) = (2, 3, 5, 7, 11 \ldots) $$ I get $$ (d^1_n) = (1, 2, 2, 4 \ldots) $$ $$ (d^2_n) = (1, 0, 2 \ldots) $$ $$ (d^3_n) = (-1, 2 \ldots) $$ $$ \ldots $$ For each sequence $(x_n)$, one can reconstruct the...

Suppose that $$ x=u^{-d},\qquad y=f(u), $$ where $$ d\in \mathbb Z_{>0},\qquad f(u)\in \mathbb C((u)), \qquad y=o(x). $$ Equivalently, $\operatorname{ord}_u f(u)>-d$. Let $ (P,Q)\in \operatorname{Aut}_{\mathbb C}\mathbb C[x,y]$ be a polynomial coordinate system, and write $$ X=P(x,y),\qquad...