29 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…

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.…

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 $i+1$.

For a smooth canonical curve, its Jacobian has a theta divisor whose double points have tangent cones, viewed as quadrics in the canonical space, and the canonical curve has an ide…

Let $C$ be the canonical curve and let $|F_4|$ denote the linear subsystem spanned by the quartic tangent cones $F_W$. The base locus of $|F_4|$ is the canonical curve $C$. Base-lo…

Let $X$ be the ambient variety and let $\Gamma$ be a set of points with ideals $I_X$ and $I_\Gamma$. Write $b_{i,j}$ for the relevant graded Betti numbers in the minimal free resol…

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…

Let $k$ be an algebraically closed field of characteristic zero, let $g\geq 3$, and let $X\subseteq \mathbb{P}_k^{g-1}$ be the canonical nodal curve obtained from $2g$ distinct poi…

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 $X$ be the relevant limiting curve, let $C_0$ be its distinguished component, let $C_i$ for $i>0$ be the other components, let $D_d$ denote the divisors indexed by $d$, and let…

Let $M_g$ be the moduli space of smooth genus-$g$ curves, let the hyperelliptic subscheme be the center of the blowup of $M_g$, and let the ribbon subscheme be the center of the bl…

Let the ribbon locus be the closure over $\mathbb{Z}[1/2]$ in the canonical Hilbert scheme, and consider the blowup of the $\operatorname{PGL}_g$-bundle over $M_g$ in the hyperelli…

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…

For a general canonically embedded curve of genus $g\leq15$ over a finite field of characteristic $p$, write $m=\lceil(g-2)/2\rceil$ and consider the critical Betti number…

Let $C\subset \mathbb{P}^{g-1}$ be a canonically embedded curve over an algebraically closed field, with coordinate ring $S_C$ and graded Betti numbers $\beta_{i,j}(C)$. Define its…

Let $C\subset \mathbb{P}^{g-1}$ be a smooth canonically embedded curve of genus $g\neq 6$, and let $k<\lceil g/2\rceil$. Denote by $W^1_k(C)$ the scheme of degree-$k$ pencils on…

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…

Rank-one locus conjecture. Only in the extremal cases $d=g-1$ and $\rho=1$ is the rank-one locus of $\chi_s$ a rational normal curve for a general curve $C$ and a general 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 $C\subset\mathbb{P}^{g-1}$ be a non-hyperelliptic canonical curve of genus $g\geq 8$ with Clifford index greater than $2$, and let $\mathcal{I}_C$ be its ideal sheaf. Wahl's va…