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One of the invariants of the fundamental group of the complement of a plane curve is the Alexander polynomial, and this invariant is easier to access than the fundamental group itself. This invariant …

A quartic surface containing a line or a conic has Picard number at least 2. There are explicit examples of Ronald van Luijk of quartic surfaces defined over $\mathbb{Q}$ such that the (geometric) Pic …

This is probably more complicated then necessary, but anyway: Let $C_1$ be the smooth genus 2 curve with affine equation $y^2=x^6+1$ and $C_2$ be the genus 10 curve with affine equation $w^6=z^6+1$. …

First concerning your question: most people use $\operatorname{Pic}(C)$ for Cartier divisors. I interpret your question as follows: you want to determine whether every divisor on $C$ is linear equival …

I encountered a similar problem a few years ago, but then in dimension 3. In that case a master student wanted to calculate the hodge diamond of a threefold which was a hypersurface $W$ in a $P^2$-bu …

For simplicity I assume that your surface $X$ is smooth and projective. For $d>3$ the following strategy often works: First of all you compute the (geometric) Picard number rank $Pic(X_{\overline{\m …

A complete intersection $X$ in $\mathbb{P}^n$ is $\mathbb{Q}$ factorial if $\dim X_{sing}<\dim X-3$. In general being $\mathbb{Q}$-factorial depends on the type of singularity you have, but also on t …

Let $C_1$ and $C_2$ be two general cubic plane curves. Let $S$ be the surface obtained by blwoing-up the nine base points. Then $S$ is a rational elliptic surface. Every section of the elliptic fibrat …

Let $\overline{S}\subset \mathbb{P}^2$ be a singular surface. Then by the Lefschetz hyperplane theorem we have that $h^1(\overline{S})=0$, but $h^3(\overline{S})$ can be nonzero, and surfaces with thi …

For surfaces in $\mathbb{P}^3$ of degree at most 3 the calculation of $Pic(X)$ is relatively easy: In this case $X$ is rational, hence $NS(X)$ modulo torsion equals $H^2(X,\mathbb{Z})$. (If you work …

Let $C\subset \mathbb{P}^2$ be a reduced nodal complex plane curve of degree $d$. Let $\Sigma$ be the set of nodes of $C$, and let $I$ be the ideal of $\Sigma$. Denote with $S=\mathbb{C}[x,y,z]$ the p …

Griffiths (On the periods of certain rational integrals. I, II. Ann. of Math. 90 (1969), 460-495 & 90 (1969), 496–541.) gave a procedure to calculate the Hodge numbers of Y. Let $S=\mathbf{C}[x_0,\dot …

You can mimic the quadric cone construction (if I did not make any mistakes in my computation). An $A_{2k-1}$ singularity is the vertex of the cone $S$ given by $x^2+y^2+z^{2k}=0$ in the weighted pro …

This question is highly non-trivial. The usual strategy is too determine the N\'eron-Severi lattice of $X$, determine all effective -2 curves and then determine all possible divisors $F$ consisting of …

In most papers on elliptic surfaces (at least among those I am aware of) the condition "there is at least one singular fiber" is to exclude elliptic surfaces that are a product or to enforce similar p …