9 problems

Let $C$ be a cuspidal plane algebraic curve and let $H_2$ be its 2-Hessian curve. At a point $p\in C$, write $m$ for the multiplicity of $C$ at $p$, $\delta$ for its delta invarian…

Let $(C,p_i)_{i=1}^\nu$ be a collection of locally irreducible local plane curve singularities satisfying … for some integer $d$. Let $c_l$ be the coefficients associated with the…

Let $C:f=0$ be a plane curve, and let $Jf e S$ be the Jacobian ideal generated by the partial derivatives of $f$. Let $If$ be the saturation of $Jf$ with respect to the maximal ide…

Rational cuspidal freeness conjecture. (i) Any rational cuspidal curve $C$ in the plane is either free or nearly free. (ii) An irreducible plane curve $C$ which is either free or n…

Let $C$ be any of the rational cuspidal curves described in Proposition 2(i), Proposition 3, or Proposition 4, with degree $d$. Freeness conjecture. If $d\geq 5$, then $C$ is a fre…

Let $C$ be a rational cuspidal curve of degree $d$ with critical points $z_1,\ldots,z_n$. Let $K_1,\ldots,K_n$ be the corresponding links of singular points, with Alexander polynom…

Let $\bar E\subseteq\mathbb{P}^2$ be a rational cuspidal curve, and let $(X,D)\to(\mathbb{P}^2,\bar E)$ be the minimal log resolution of singularities. Zaidenberg's finiteness conj…

Let $\bar E\subseteq\mathbb{P}^2$ be a rational cuspidal curve of log general type, and let $\pi:(X,D)\to(\mathbb{P}^2,\bar E)$ be the minimal log resolution of singularities. Let…

Let $C\subset \mathbb{P}^2$ be a plane rational cuspidal curve, meaning a rational plane curve whose singularities are cusps. Orevkov's four-cusp conjecture. The curve $C$ can not…