55 problems

Let $\mathrm{OG}$ be the odd quadric considered above, let $c_1=c_1(-K)\in H^2(\mathrm{OG})$ be its anticanonical class, and let $\operatorname{FPdim}(A(c_1))$ denote the Frobenius…

Let $X=Q_1\cap Q_2\subset\mathbb{P}^{n+2}$ be a smooth complete intersection of two quadrics in an odd-dimensional projective space. Let $C$ be the hyperelliptic curve associated w…

Let $k$ be a number field and let $X\subset \mathbb{P}^n_k$, with $n\geq 6$, be a non-conical geometrically integral intersection of two quadrics. Suppose that $X$ has a smooth…

For fixed $d\geq 1$ and each prime $p>2$, let … and let $\operatorname{e_{HK}}(A_{p,d})$ be its Hilbert–Kunz multiplicity. Yoshida's conjecture. The sequence…

Let $Q_n\subset\mathbb P^{n+1}$ be a smooth quadric hypersurface of dimension $n$ over an algebraically closed field. Property (D) is the assertion that the diagonal…

Let $q$ be an anisotropic quadratic form over a field $k$. Its first Witt index is the Witt index of $q$ over the function field of its associated projective quadric. Converse to K…

Let $Q$ be the quadric under discussion, with spinor bundles $\Sigma^+$ and $\Sigma^-$. Consider the strong complete exceptional sequence … A braid-orbit conjecture for the quadric…

Let $1 \le k \le n-2$. For smooth quadrics $Q_1,\ldots,Q_{d_{k,n}} \subset \mathbb{P}_{\mathbb{R}}^n$, let $\operatorname{sign}(Q_i)$ denote the signature of $Q_i$, and let…

Extremal-curve conjecture. The extremal curves should be special curves, so that the correct upper bound for the genus should be the bound given in (3.5.1).

Let $M\subset\mathbb R^n$, $n\geq 3$, be an open patch of a hypersurface of class $C^2$, and let $\Gamma_1,\Gamma_2$ be space curves. Orthogonal-intersection conjecture. If for eve…

Let $\operatorname{DQM}_{\operatorname{gm}}(F;\mathbb{Z})$ denote the category of geometric motives generated by smooth projective quadrics over a real closed field $F$ with integr…

Let $\langle Q_1,Q_2\rangle$ be a pencil of quadrics in $\mathbb{P}^{n+2}$ whose base locus is smooth in an odd-dimensional projective space. Let $C$ be the hyperelliptic curve ass…

Let $X$ be a projective scheme and let $L$ be a sufficiently ample line bundle on $X$. A quadric of rank $3$ means a degree-two equation whose associated quadratic form has rank…

For $n\in\{2,4,8,16\}$, let $W_{n,\operatorname{sym}}$ be a linear subspace of $\mathbb{R}_{\operatorname{sym}}^{n\times n}$ with … and suppose every nonzero element of…

Let $n\in\{8,16\}$, and let $L\subset\mathbb{R}_{sym}^{n\times n}$ be a subspace satisfying … and such that $\boldsymbol{\mathrm{P}}(L^\vee)\cap\boldsymbol{\mathrm{Z}}$ exists and…

Let $M$ be a symmetric convex body, and let $x\notin M$. Say that $M$ looks symmetric from $x$ when its graze from $x$ has the corresponding symmetry property. Bianchi–Gruber conje…

Let $M,K$ be convex bodies in $\mathbb{R}^n$ with $M\subset\operatorname{int}K$ and $n\geq 3$. For $x\in\operatorname{bd}K$, let the graze $\Sigma(M,x)$ be the set of contact point…

Let $\mathcal M\subset\mathbb R^{n+1}$ be a connected smooth hypersurface. Suppose that … for every $x\in\mathcal M$ and every sufficiently small $r>0$. Kowalski–Preiss conjecture.…

Let $X$ be the curve considered in the case $p=e_2$, and let $\delta(X,k)$ denote the delta invariant associated with rank-$k$ quadratic forms. Delta-gap conjecture. … The argument…

Let $q_n$ denote the quadratic form in $n$ variables and let $\operatorname{rk}(q_n^s)$ be its Waring rank. For fixed $n\in\mathbb{N}$, the rank is compared with the generic rank o…

Let $q$ be the quadric and let $N(n,r)=\binom{n+r}{n}$ be the cactus rank of $q^r$. Denote by $\operatorname{VAPS}_G(q^r,N(n,r))$ the variety of smoothable apolar schemes of length…

Let $\mathbb{K}$ be an algebraically closed field, let $N\geq 2$ and $d\geq 2$ be fixed integers, and let $\alpha=(\alpha_{ij})$ be an $N\times N$ skew-symmetric matrix with entrie…

Let $H$ be a hyperplane of ${\rm PG}(r+1,q)$, and let an affine vector space partition be a set of subspaces not contained in $H$ such that every point outside $H$ lies in exactly…

Let $H_{\infty}$ be a hyperplane at infinity of ${\rm PG}(6,q)$, and let $\mathcal Q$ be a hyperbolic quadric in $H_{\infty}$. A hyperbolic affine spread is an affine vector space…

Let $q$ be a prime power greater than $2$, and let $\mathcal{Q}(8,q)$ be the parabolic quadric in projective $8$-space. An $(q^2+q+1)$-ovoid is a set of points meeting every genera…