16 problems

Generic Green's conjecture. The generic curve $C$ of genus $g$ satisfies

Generic geometric syzygy conjecture. For every $p$, the space of $p$-th linear syzygies of $C$ is spanned by scrollar syzygies.

Let $C \subset \mathbb{P}^{g-1}$ be a general canonical curve of genus $g$. A scrollar syzygy is a $k$th syzygy of rank $k+2$, and the corresponding schemes of such syzygies are ca…

Geometric Syzygy Conjecture. All linear syzygy spaces $\mathrm{K}_{i,1}(C,\omega_C)$ are spanned by syzygies of minimal rank $i+1$.

Let $C\hookrightarrow\mathbb{P}^{g-1}$ be a generic canonical curve of genus $g\geq 9$, and let $p\in\mathbb{P}^{g-1}$ be a generic point. Write $C'$ for the projection of $C$ away…

Let $C$ be a smooth complex projective curve of genus $g\geq 2$, let $C_k$ denote its $k$-th symmetric power, and let $K_C$ be the canonical bundle. Say that a line bundle $L$ on…

The Geometric Syzygy Conjecture in Even Genus. The last linear syzygy space

Folk conjecture. The tangent developable surface $T$ satisfies Property $(N_p)$ for

Let $X\subseteq \mathbf{P}^{g-1}$ be a non-hyperelliptic canonical curve of genus $g$. Its Clifford index is … Here $r(\varphi)$ is the dimension of the linear span of the image an…

Let $C\subset\mathbb{P}^{g-1}$ be a general canonical curve of genus $g$, let $k$ be a positive integer with Brill–Noether number $\rho(g,k,1)\geq0$, and let $g^1_k$ be a general p…

Let $k$ be an integer and let $R_{2k+1}$ be the balanced canonical ribbon of genus $2k+1$. For integers $p$ satisfying $2\leq p\leq k-1$, consider its $(p,2)$-syzygy point…

Let $C\subset \mathbb{P}^{g-1}$ be a generic canonical curve. For integers $p$ with $p\geq \lfloor (g-3)/2\rfloor$, consider its $(p,2)$-syzygy point…

Let $C$ be a general non-hyperelliptic smooth projective curve of genus $g\geq 7$, canonically embedded as $C\hookrightarrow\mathbb{P}^{g-1}$, and let…

Let $C$ be a general canonical curve of genus $g\geq 7$, embedded by its canonical linear system in ${\mathbf P}^{g-1}$, and let $N_{C/{\mathbf P}^{g-1}}$ denote its normal bundle.…

Let $C$ be a smooth bicanonical curve of genus $g\geq 3$, and let $m^{th}$ Hilbert point denote the Hilbert point associated to its bicanonical embedding at level $m$. Morrison's c…

Farkas's conjecture. A general smooth normal curve $C\subset\mathbb P^r$ of odd genus $g=2p+3$ and degree $d=2g$ has a pure resolution, equivalently satisfies $N_p$. A general smoo…