Let $X$ be a smooth, projective complex algebraic variety. Let $D$ be a simple normal crossings divisor on $X$, with irreducible components $D_i$, for $i \in I$. For each non-empty …

I would like to know if the following statement is correct. Statement. Let $X$ be a normal projective variety with $Pic(X)=\mathbb Z+torsion$. Let $L$ be an ample line bundle on $ …

Let $X$ be a normal complex projective variety, let $V$ be a closed subset of $X$ (possibly reducible), and let $I_V$ be its ideal sheaf (consider the reduced scheme structure for …

Assume we are given two smooth curves $C_1$ and $C_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms b …

Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$. $E$ a vector bundle over $X$ with a divisor $D$. We look at the sections $A:=H^{0}(X,E)$ with …

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes les …

Let $X \hookrightarrow \mathbb{P}^n$ be a hypersurface of degree $d$. I am trying to prove that $\mathcal{O}_{\mathbb{P}^n}(X)=\mathcal{O}(d)$. My idea is the following: if one con …

Let $D$ be a simple normal crossing divisor on a smooth projective variety over a field $k \subset \mathcal{C}.$ Write $D_i$ with $i \in I$ for its irreducible components. Denote, …

I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities". Let $X$ be a nonsingular projective variety over $\mathbf C$ and …

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this …

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ an …

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions …

Consider an infinite set $S$, of positive integers. If all the finite subsets of $S$ have GCD $>$ $1$, is it necessary that the GCD of $S$ is greater than $1$ as well?

Let $S$ be a smooth affine variety over an algebraically closed field (this could be the field of complex numbers). Is there an 'easy' way to verify whether $NS(S)=0$? Unfortunate …

Let $k$ be a fixed algebraically closed field and $X/k$ an irreducible scheme smooth and proper over $k$. Can there exist a line bundles $\mathcal{L}, \mathcal{M}$ and an integer $ …