MathOverflow

I am not well versed in algebraic geometry but am trying to learn about certain linear systems over the 2nd Hirzebruch surface. I would appreciate any references to where I can learn about this kind of work in an approachable way or if an answer to my...

In the moduli space of elliptic curves over $\mathbb{C}$, the two most symmetric points with Complex Multiplication (CM) correspond to the $j$-invariants $j=0$ and $j=1728$.
$j=0$: Corresponds to the hexagonal lattice $\Lambda_{\omega} = \mathbb{Z} \oplus \mathbb{Z}\omega$ (where $\omega = e^{i\pi/3}$), with an automorphism group of order 6....

A complex polynomial $\mathbb{C}[x,\bar{x}]$ is called Hermitian if it satisfies $$p(x, \bar{x}) = \overline{p(x, \bar{x})}.$$
This means that the polynomial outputs real values. The polynomial can be expanded into the monomial basis.$$p(x, \bar{x}) = \sum_{\alpha, \beta \in \mathbb{N}^d} c_{\alpha, \beta} x^{\alpha} \bar{x}^{\beta}$$...

For any positive integer $k$, let us define $S(k)$ as the set of $x_1+\ldots+x_k$with $x_1,\ldots,x_k$ positive integers such that $[x_1,\ldots,x_k]$ (the least common multiple of $x_1,\ldots,x_k$) is a product of $k$ consecutive integers.
Questions. Let $k>2$ be an integer. Is it true that every integer $n\ge 2^k-2$ belongs to the set $S(k)$? Is there a positive integer $N(k)$ such that $S(k)=\{n\ge N(k):\ n\in\mathbb Z\}$?...