9 problems

Let $m$ be the highest multiplicity among the singular points of a rational cuspidal plane curve of degree $d$, and let $\alpha=(3+\sqrt{5})/2$. Orevkov constructed two families of…

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 $(C,p_i)_{i=1}^\nu$ be a collection of locally irreducible plane curve singularities, and let $\Delta_i(t)$ be their monodromy characteristic polynomials. Put…

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…

Let $C:f=0$ be a reduced plane curve. Suppose that $C$ is rational and cuspidal, meaning that it is rational and all its singularities are cusps. A curve is free or nearly free acc…

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…

Negativity conjecture. One has $\kappa_{1/2}(\mathbb{P}^2,\bar E)=-\infty$. Equivalently, $h^0(m(2K_X+D))=0$ for every $m>0$. This conjecture predicts the vanishing of all positive…

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…