25 problems

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,1}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the Koszul cohomo…

Complete intersection classification conjecture. Such a subvariety exists if and only if it is one of the following:

For each degree, choose the lex-leading monomial and use it to produce nonzero Betti entries in the corresponding row of the Betti table. Ein–Erman–Lazarsfeld conjecture. Choosing…

Let $M$ be a Veronese module, let $\beta_{i,j}(M)$ denote its graded Betti numbers, and consider the cited range of entries in a given row. Ein–Erman–Lazarsfeld conjecture. All Bet…

Let $M$ be the coordinate ring of the Veronese, under the hypotheses of Theorem, and let $\beta_{i,i+1}$ denote the corresponding graded Betti numbers. Erman–Martinova conjecture.…

Ulrich rank sufficiency conjecture. The condition $6\mid r(a^2-1)$ should be sufficient for the existence of a rank $r$ Ulrich bundle on…

Positive geometry conjecture. The closure

For integers $d,n\geq 1$, set … and let $X_d^n\subset\mathbb{P}^{N_d^n}$ be the $d$-th Veronese $n$-fold; write $X_d=X_d^3$ when $n=3$. For $d\geq 2$, let $X_d$ be the $d$-th Veron…

Ein–Lazarsfeld's conjecture. These inequalities are equalities whenever $d\geq b+q+1$. This conjecture concerns the precise first and last nonvanishing positions of Veronese syzygi…

The three-dimensional restriction conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if, for all $i<j<k$, the algebra…

Maximal-loss conjecture. In each projection step, one loses the maximal number of quadrics, namely the codimension of the projected variety. Equivalently, successive general-point…

Planar Veronese secant-index conjecture. The maximal number of points in the finite reduced intersection of $r$ linearly independent degree-$d$ curves is $b_{r-1}$. The statement p…

Let $v_d^{(n)}(\mathbb{P}^n)$ be the degree-$d$ Veronese variety, let $\mathfrak{L}(X)$ be its sequence of secant indices, and let ${\mathfrak{R}\mathfrak{L}}^G(X)$ and…

Quadratic generation conjecture. The ideal defining $\mathcal{R}_{d,k,m}$ is generated by quadratic polynomials.

Let $k$ be a field, let $F\in k[x_0,\dots,x_n]_d$ be a homogeneous polynomial, and let $\mathcal{V}_d^n$ be the degree-$d$ Veronese variety diagonally embedded in the Segre variety…

Let ${\rm Ver}(d-1,k)$ be the projective toric Veronese variety defined by the matrix whose columns are the non-negative integer vectors with coordinate sum at most $k$, where…

Precise nonvanishing range conjecture. In the situation of that theorem, one has

Let $s,d,n$ be positive integers, let $\nu_d\mathbb{P}^n$ be the degree-$d$ Veronese re-embedding of projective space, and write $\sigma_s(\nu_d\mathbb{P}^n)$ for its $s$-secant va…

Catalisano–Geramita–Gimigliano conjecture. The secant variety $\sigma_s({\mathcal T}_{n,d})$ is always nondefective, except in the exceptional cases listed in Theorem.

Let $X=v_d(\mathbb P^n)$ be the $d$th Veronese embedding of projective $n$-space, and define … Here $g_j(X)$ denotes the $j$th component of the gap vector of $X$, and…

Let $S^{(d)}$ be the $d$-th Veronese ring of a polynomial ring, and let its minimal graded free resolution have homological degree indexed by $i$. A resolution is linear through ho…

Let $K$ be a field, let $n,d,ke\mathbb{Z}$ with $k,n\geq 2$ and $d\geq 2k-2$, and let $Cat(t,d-t;n)$ denote the $t$-th generic catalecticant. Write $I_k(Cat(t,d-t;n))$ for the idea…

Let $v_k(\mathbb{P}^d)$ be the $k$-uple Veronese embedding of $\mathbb{P}^d$, with $d\geq2$ and $k\geq3$, and let $\Sigma$ be its secant variety. Write $N_{p,q}$ for the indicated…

Let $v_d(\mathbb P V)\vartriangleleft\mathbb P S^dV$ be the Veronese variety, and let $\sigma_2(v_{d+e}(\mathbb P^n))$ be its second secant variety. The determinantal ideal from th…

Let $V$ be a vector space, let $k>2$, and let $X=v_k(\mathbb{P}V)$ be the Veronese variety. For a positive integer $r$, define … Let $R_S$ be the minimum $r$ such that $Y_r=S^kV$.…