53 problems

Equal matroids conjecture. The two join varieties have the same algebraic matroid:

Let $\mathcal{C}(E,D,X)$ be the toric code associated to the $1$-cycle $E$, the $T$-invariant Cartier divisor $D$, and the toric variety $X$. Assume that $X$ is a nonsingular toric…

Let $I$ be a toric ideal defining a projectively normal $r$-dimensional toric variety. The normality Gröbner-basis conjecture. The ideal $I$ has a Gröbner basis consisting of binom…

The $G^1$-degeneration conjecture. The spectral sequence $G^k_{p,q}$ for $H_*(X(\mathbb{R}))$ degenerates at the $G^1$ level.

A real algebraic variety is a complex variety $X$ defined over the real numbers, with real part $X(\mathbb{R})$ equal to the fixed-point set of complex conjugation. It is an M-vari…

Let $n$ be a positive integer. Define the commutative ring $\mathbf{M}^{\text{def}}(n)$ generated by variables $\mathsf{S}_i$ for $i\in[n]$ and $\mathsf{S}_\alpha$ for…

Toric-code degree conjecture. If $q$ is sufficiently large, then every $f\in L(G)=H^0(X,\operatorname{\mathcal O}(G))$ has at most $n=\operatorname{deg}(P)$ zeros in…

Let $d$ be the parameter in Hansen's theorem and let its lower estimate be the stated inequality involving $d$. The equality-case conjecture. Hansen's theorem continues to hold whe…

Let $M$ be an arbitrary commutative, cancellative, torsion-free monoid without non-trivial units, let $R$ be a regular ring, and let $i>0$. For every sequence…

Let $X$ be a smooth proper toric variety, and consider its Thomsen stratification and the associated intersection numbers. Thomsen stratification intersection-number conjecture. Fo…

Let $X$ be a projective toric variety of dimension $n\geq 2$, let $L$ be a line bundle on $X$, and let $p\geq 0$. Property $N_p$ is the syzygy property defined by the minimal grade…

Let $X_\Sigma$ be the toric variety and let $f_1,\ldots,f_k$ satisfy Assumption. Let $Y\subseteq\mathbb{C}^n$ be the affine variety constructed from these polynomials as above. Its…

Monotonicity and vanishing conjecture. The function $f(d)$ is strictly monotone decreasing for $d>1$, and

Let $\Sigma$ be a rational fan defining a projective toric variety $X_\Sigma$. The combinatorial intersection cohomology is denoted $IH_{\mathrm{comb}}^*(\Sigma,\mathbb{Q})$, and f…

Let $X$ be a smooth projective variety, let $Z$ be a complete toric variety, and let … be a surjective map. Occhetta–Wiśniewski's conjecture. Then $X$ is a toric variety. The paper…

Achinger–Witaszek–Zdanowicz conjecture. There exists a finite étale Galois cover $f:Y\to X$ such that the Albanese morphism

Let $G$ be a semisimple algebraic group, let $P_{\lambda}$ be the parabolic subgroup associated with a dominant weight $\lambda$, and let $G/P_{\lambda}$ be embedded by the corresp…

Let $(\mathcal{D}^{\star}_X,\mathcal{D}_X)$ be a transpolar pair of VEX multitopes. A center-enclosing sub-simplex reduction is a pair of lattice simplices…

A toric variety is an algebraic variety equipped with the torus action arising from a toric construction; a Calabi–Yau hypersurface is a hypersurface with trivial canonical class.…

Toric integral-model Hilbert property conjecture. If $N_M=N$, then $(\mathcal{X},\mathcal{M})$ satisfies the $\mathcal{M}$-Hilbert property over every open $B\subsetneq C$. If furt…

Let $w$ and $w'$ be scalings for the Segre embedding of $\mathbb P^1\times\mathbb P^1\times\mathbb P^{n}$, and let $P_n$ denote the relevant polytope. For every face…

Let $\mathcal T$ be a staged tree with non-toric $V(\ker \varphi_{\mathcal T})$ that uses color set $S$. Let $\mathcal T'$ be a staged tree such that its restriction to color set…

Let $X$ be a smooth projective variety over an algebraically closed field. For a coherent sheaf $E$ on $X$ and a point $p\in X$, write $\varepsilon(E,p)$ for its Seshadri constant…

Let $Y$ be a toral variety, and suppose that its maximal torus in $\operatorname{Aut}(Y)$ is trivial. Finiteness question. Is $\operatorname{Aut}(Y)$ a finite group? This is posed…

Berkesch–Erman–Smith's conjecture. The module $M$ admits a virtual resolution of length at most