26 problems

Let $\phi(n,d)$ denote the polynomial that gives the dimension of the space of quadrics in $n$ variables passing through $d$ general points and tangent to $\binom{n+1}{2}-d-1$ gene…

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 $M\subset\mathbb R^n$, $n\geq 3$, 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 $\langle Q_1,Q_2\rangle$ 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 $X=Q_1\cap Q_2\subset\mathbb{P}^{n+2}$ be a smooth complete intersection of two quadrics in an odd-dimensional projective space. Let $C$ be the hyperelliptic curve associated w…

Let $X$ 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 $M,K$ 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 $q$ 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 $q$ 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…

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…

Real critical points conjecture. There exists $s=(s_1,\ldots,s_n)\in\mathbb{R}^n$ such that $\widetilde{\ell}_S$ has $2r-3$ distinct real critical points.

Let $X$ be an arc of size $2k+3$ in $\mathrm{PG}(k-1,{\mathbb F})$, and let $U$ be the subspace of quadratic forms that vanish on $X$. If … then projection conjecture. The projecti…

Let $\varphi$ be an anisotropic quadratic form over a field $F$, let $\mathrm{dim}\varphi\geq 2$, let $\frak{i}_{1}(\varphi)$ denote its first Witt index, and for a nonzero integer…

Let $F$ be a field of characteristic different from $2$, let $\varphi$ be an anisotropic quadratic form over $F$, and let $F(\varphi)$ be the function field of its projective quadr…

An elliptic quadric is a cap in $\mathrm{PG}(3,q)$, and $\mathrm{PG}(3,q)$ can be partitioned into $q+1$ disjoint elliptic quadrics. The set of elliptic quadrics induces factors on…

Let $\gamma$ be a net of quadrics, meaning a three-dimensional linear system of quadratic forms, and suppose that $\gamma$ is semistable. Semistable-net conjecture. If $\gamma$ is…

Let $X$ be an anisotropic quadric over a field $k$. Write $\mathfrak{i}_1(X)$ for its first Witt index. A subquadric $X'\subset X$ is a quadric contained in $X$. Totaro's birationa…

Let $X$ be an anisotropic quadric over a field $k$. Write $\mathfrak{i}_1(X)$ for its first Witt index, and say that $X$ is ruled if it is birational to $Y\times\mathbb{P}_k^1$ for…

Let $X$ and $Y$ be anisotropic quadrics of the same dimension over a field $k$. A rational morphism means a rational map between these varieties, defined on a dense open subset. Qu…

Let $X \to \mathbb{P}^1$ be a fibration in complete intersections of two four-dimensional quadrics over an algebraically closed field of characteristic zero. Write $\mathcal{A}_X$…

Let $R/I$ be an artinian Gorenstein algebra of codimension $r\geq 3$ and socle degree $e$, and assume that $h_2=r$. The $h_2=r$ conjecture. Then … Furthermore, if $e\geq 4$, the id…

Let $R/I$ be an artinian Gorenstein algebra presented by quadrics, with socle degree at least $3$, and suppose that $R$ is defined over a field of characteristic zero. The WLP conj…

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $S$ be a polynomial ring over $\Bbbk$, and let $I\subseteq S$ be a prime ideal containing no linear form. S…