31 problems

Let $X\subset\mathbb P^2$ be a rational cuspidal plane curve, meaning a rational plane curve whose singularities are cusps. Dimca–Sticlaru conjecture. Every rational cuspidal plane…

Let $D$ be a free divisor in a complex manifold $X$. The logarithmic comparison conjecture. If the logarithmic comparison theorem holds for $D$, then $D$ is strongly Euler homogene…

Let $M$ be a complex manifold and let $D$ be a reduced free divisor of $M$. An irreducible subvariety $Z$ is weakly log transverse to $D$ when it satisfies the paper's weak log tra…

Let $M$ be a complex manifold, let $D\subset M$ be a free divisor, and suppose that $D$ satisfies the logarithmic comparison theorem. Let $M$ be regarded as the ambient constructib…

Let $M$ be a complex manifold and let $D$ be a reduced free divisor of $M$. A constructible function $\gamma$ is weakly log transverse to $D$ when it satisfies the weak log transve…

Let $X$ be a complex analytic manifold of dimension $n$, and let $D\subset X$ be a free divisor. Say that $D$ satisfies LCT when the inclusion of logarithmic into meromorphic de Rh…

Plane-curve basis conjecture. There always exists a basis of the logarithmic derivations of $D$ such that one basis element has no linear part.

LCT–strong Euler-homogeneity conjecture. If $D$ satisfies the Logarithmic Comparison Theorem, then $D$ is strongly Euler-homogeneous.

Let $D\subset X=\mathbb{C}^{4n-10}$ be the linear free divisor $D_n$. The Hodge filtration on $\mathcal{O}_X(*D)$ is generated at level $k$ when … for every $l\geq 0$. Generating-l…

Let $X$ be a complex manifold, let $D$ be a divisor satisfying the assumptions of Theorem and Corollary, and let $b_D(s)$ be the relevant Bernstein–Sato polynomial. For a well-filt…

Freeness conjecture for ${\mathcal F}{\mathcal Z}_2$. The curves $C_k\in {\mathcal F}{\mathcal Z}_2(k)$ are free divisors and

Freeness conjecture for ${\mathcal F}{\mathcal Z}_1$. The curves $C_{d,a}\in {\mathcal F}{\mathcal Z}_1(d,a)$ are free divisors and $mdr(f)=2$ for all possible values of $(d,a)$ wh…

Walther-bound equality conjecture.

Let $C$ be a rational cuspidal curve, meaning a rational plane curve whose singularities are cusps, and suppose that $C$ is not a free divisor. The authors' conjecture. The curve…

Let $k$ be a field, let $f$ define a reduced free divisor that is a line arrangement in ${\mathbb P}_k^2$, and set $X=V(f)$. Let $J(f)$ be the gradient ideal of $f$, and let $I$ be…

Let $f$ be the defining polynomial of either an arrangement ${\mathcal A}:f=0$ of $d$ hyperplanes in $\mathbb{P}^n$, or a free locally quasi-homogeneous divisor $V:f=0$ of degree…

Faber's conjecture. A free divisor has only normal crossing singularities if and only if it is normal crossing in codimension $1$.

Let $C\subset\mathbb{P}^2$ be a rational cuspidal curve. A plane curve is free or nearly free according to the definitions of free and nearly free divisors, respectively. Free-or-n…

Branch-number conjecture. (i) Any free irreducible plane curve $C$ has only singularities with at most two branches. (ii) Any nearly free irreducible plane curve $C$ has only singu…

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 $C:f=0$ be a plane curve of degree $d$. Let $ct(f)$ and $st(f)$ denote the coincidence and stability thresholds of its Milnor algebra, and let $T=3(d-2)$. Freeness criterion co…

Let $C\subset\mathbb{P}^2$ be an irreducible plane curve of degree $d\geq 2$, and suppose that $C$ is a free divisor, meaning that its logarithmic vector-field bundle splits as a d…

Let $hh_1^{r_1}\cdots h_s^{r_s}=0$ be the equation of the unique non-reduced curve, where ${h=0}$ is reduced, $\operatorname{deg}(h_i)=m_i\geq 1$, and $r_i\geq 2$. Let $D_k$ be a m…

Let $D \subseteq S$ be a divisor in a complex manifold $S$, defined at a point $p$ by $h \in \mathcal{O}_{S,p}$. Assume that its Jacobian ideal $J_h$ is radical, equidimensional, a…