Let $C$ be a plane curve and suppose, for simplicity, it has a singularity at the origin. Let $\tilde{C}$ be it's strict transform after blowing up $(0,0).$ Is it possible for $\tilde{C}$ to have a singular point not at the origin?

These days I've been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on infinite root stacks. At the moment I'm looking at their proposition 4.9....

I mean, if a function $F(s)$ is in Selberg class, and $F(s)$ has a pole of order m at $s=1$, is it true that there exists a function $G(s)$ in Selberg class such that $F(s)=\zeta^m(s)G(s)$, and $G$ is entire?

Suppose $R$ is a reduced Noetherian ring. We know that $f \in R$ is a zero divisor if and only if $V(f)$ contains an irreducible component of $\mathrm{Spec} R$. I would like to know if one could use the irreducible components to say something about the annihilator of $f$ in $R$? (Perhaps it...

I found this problem where I think there is a mistake but am not 100% sure: Let $Z$ be a closed subset of a topological space $X$. If $Z$ is irreducible we call codim($Z,X$) as the supremum of lengths of the chains of irreducible closed subsets of $X$ which contains $Z$: $$Z\subset Z_0\subsetneq...

One of the axioms for Selberg class is that for some natural number $m$, $(s-1)^mF(s)$ extends to an entire function of finite order. What is the use of "finite order"? If we allow it to be of infinite order, what will happen?

Elliptic curves have L-functions that correspond to modular forms. Elliptic curves are degree 3 algebraic curves. I want to know if quadratic curves have L-functions. If they do, are these L-functions related to modular forms?

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...

Context While working with Ramanujan's $P(q)$ function: $$P(q)=1-24\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}, \hspace{.5cm} 0<|q|<1.$$ I have found the following evaluation: $$S=1-24\sum_{n=1}^{\infty}\frac{ne^{-2\pi n/5}i^{8n/5}}{1-e^{-2\pi n/5}i^{8n/5}}=\frac{15}{\pi},\tag{1}$$ Being...

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...