7 problems

Let $X$ be a general canonical curve of genus $g\geq7$, embedded by its canonical linear series, and let $N_{X/\mathbb{P}^{g-1}}$ be its normal bundle. Aprodu–Farkas–Ortega's conje…

A BN-triple is a triple $(g,r,d)$ specifying a Brill–Noether curve of genus $g$, degree $d$, and dimension $r$. For a general BN-curve $X\subset\mathbb{P}^r$ of type $(g,r,d)$, con…

Let $G(k,n)$ be the Grassmannian, with $k\leq n-k$, and write … where $0\leq r_1<k$ and $0\leq r_2<n-k$. Let $X$ be a general rational curve of degree $e$ in $G(k,n)$. Clarke–Larso…

Let $X$ be a smooth nondegenerate rational curve of degree $e$ in projective space $\mathbb{P}^r$, and consider the locus of such curves whose normal bundle $N_{X/\mathbb{P}^r}$ ha…

Coskun–Larson–Vogt's finiteness conjecture. There are finitely many triples $(d,g,r)$ with $g\geq 2$ for which $N_{C/\mathbb{P}^r}$ is not stable.

Let $X$ be a general quintic hypersurface in $\mathbb P^4$, and consider rational curves on $X$. Clemens' conjecture. Every rational curve on $X$ should be rigid; equivalently, it…

Let $R_n\subset\mathbb{P}^n$ be the rational normal curve, and let $X\subset\mathbb{P}^n$ be a general Fano complete intersection of $k$ quadric hypersurfaces containing $R_n$. The…