11 problems

Let $M_N^k$ be a degree-$k$ hypersurface in ${\mathbb P}^{N-1}$, let $w({\cal O}_{h^a}{\cal O}_{h^b}|\prod_{p=0}^{N-2}({\cal O}_{h^p})^{m_p})_{0,d}$ denote its genus-$0$ multi-poin…

Let $X\subset \mathbb{P}^n$ be a general hypersurface of degree $d\leq n$ and dimension at least $3$. Let $R_e(X)$ be the open locus in the Hilbert scheme of $X$ parameterizing smo…

Rational-curve family conjecture. If $d\leq n-1$ and $n>3$, then the family of rational curves of degree $e$ on $X$ is irreducible and has the expected dimension.

Tian's conjecture. Hilbert–Mumford stability of $X_f$ is sufficient for the existence of a Kähler–Einstein metric on $X_f$.

Let an infinite series of anticanonically embedded quasi smooth Fano hypersurfaces in weighted projective 4-spaces be given. The series in the known classification have the form ……

Let $M_6^k$ be a four-dimensional degree-$k$ hypersurface in $\mathbb{CP}^5$, and let $\langle({\cal O}_{h^2})^a({\cal O}_{h^3})^b({\cal O}_{h^4})^c\rangle_{g,d}$ denote the genus-…

Let $M_N^k$ be a degree-$k$ hypersurface in ${\mathbb P}^{N-1}$. Let $F_1^{N,k,A}(t^0,t^1,\ldots,t^{N-2})$ be the generating function of genus-$1$ Gromov–Witten invariants, let…

Folklore conjecture. Every smooth Fano hypersurface of dimension $4$ admits a Kähler–Einstein metric.

Coskun–Starr's conjecture. If $h\leq n$ and $n\geq 4$, then $R_d(X)$ is irreducible and has the expected dimension

Let $V_d\fieldsubset {\fieldmathbb P}$ be a smooth hypersurface of degree $d\fieldleq M$, where $d\fieldgeq [(M+5)/2]$; then $V_d$ is a Fano variety of index $r=M+2-d$. Let…

Tropical Lagrangian torus conjecture. The torus satisfies