15 problems

Barja–Stoppino slope conjecture. Under these hypotheses, $f$ satisfies the slope inequality.

Let $f:S\to B$ be a locally non-trivial fibration, and write $\omega_{S/B}$ for its relative canonical sheaf and $\lambda_f$ for its slope. A locally free quotient of a vector bund…

Let $f\colon S \longrightarrow B$ be a fibration of genus $g$, and let $q_f$ denote its relative irregularity, with $q_f<g-1$. The slope $s(f)$ is the ratio of the relevant relativ…

Let $X\to B$ be a general family of genus $g$ stable curves whose general member has Clifford index $\mathfrak{c}$, and let $B$ be a general curve in $\overline{\mathcal D}^r_d$. C…

Let $X$ be a family of $d$-gonal curves of genus $g$, with base $B$ not contained in a certain codimension-one closed subset of $\overline{\mathcal D}_d$. Semistable slope-bound co…

Let $\mathcal{F}_d(g)$ be the exact upper bound for slopes of families of stable curves with smooth $d$-gonal general member, and let $g_d=2d-3$ for $d\geq 3$ and $g_2=2$. Slope-bo…

Let $g$ be even, let $B$ be a base curve not entirely contained in the Maroni divisor, and let $X\to B$ be a family whose singular members belong only to…

Reduced-fiber-free slope equality conjecture. The slope equality

Zhang's conjecture. For every connected polarized weighted graph $\overline{G}$ of genus $g\geq 2$, the slope satisfies

For each $g\geq 2$, let … where $\lambda_f$ denotes the slope of $f$. Infimum-slope conjecture. … for all $g\geq 2$. The paper establishes that the infimum is positive and that, fo…

Higher-dimensional slope inequality. Under these hypotheses, the fibred variety satisfies the slope inequality. This is known for relatively minimal fibred surfaces of genus at lea…

Slope inequality conjecture. Under these assumptions, the fibration satisfies the displayed slope inequality. The inequality is equivalent to the $f$-positivity of $\omega_f$ and f…

Slope inequality conjecture. For every genus-$g$ Lefschetz fibration $f:X\rightarrow S^2$, the slope inequality

Moriwaki's conjectural strengthening. The divisor $M$ has non-negative degree on each complete curve in $\overline{\mathcal M}_g$ that does not lie in $\Delta_0$. The analogous two…

Let $f:X\to Y$ be a semistable fibration, let $CV(f)$ be the set of singular fibers, and let $\Gamma_y$ be the polarized metrized graph of the fiber over $y\in CV(f)$. For each nod…