22 problems

Integral-closedness conjecture. If $\mathcal{R}'(I)$ is nearly Gorenstein, then $I$ is integrally closed.

Let $G$ be a finite simple graph, and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring associated to $G$. Write $\mathcal{R}(I_c(G))$ for the Rees algebra of…

Hara–Watanabe–Yoshida conjecture. If $T$ is $F$-rational, then $S$ is $F$-rational.

Fiber-type classification conjecture. (1) If

Packing-property normality conjecture. The Rees algebra $S[Iz]$ is normal.

Simis' reformulation. The map $F$ is a Cremona map if and only if the Rees algebra $S[Iz]$ is normal.

Simis' conjecture. If $A$ has entries in $\{0,1\}$ and $\det(A)=\pm d$, then the Rees algebra $S[Iz]$ is normal.

Let $G$ be a graph, let $J_G$ be its binomial edge ideal, and say that $J_G$ is of linear type when its Rees algebra is generated by the degree-one part in the standard sense. Line…

Let $I_n(B)$ be the ideal associated with the matrix $B$ in the construction of the Rees algebra, and let $\mathcal{P}$ denote the non-linear part of the defining ideal of that Ree…

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, let $I=(f_1,\dots,f_r)$ be a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal of $S$, and let $\Theta$ be the Jacobian…

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, let $I=(f_1,\dots,f_r)$ be a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal of $S$, and let … be the rational map def…

Let $D$ be a digraph on $n$ vertices, let $G(D)$ be its associated bipartite graph, and let $M_D$ be the perfect matching associated with $D$. If $J$ is the defining ideal of the R…

Let $G$ be a finite simple graph, let $J_G$ denote its binomial edge ideal, and let $C_n$ denote the cycle graph on $n$ vertices. Let $\mathcal{R}(J_G)$ be the Rees algebra of…

Let $d=2s+1\geq 7$. Given three general forms $f_1,f_2,f_3$ of degree $d$ in $R$, let $I=(f_1,f_2,f_3)$, let $\delta(I)$ be the ideal generated by the associated degree-$d-2$ forms…

Let $I$ be the ideal considered in the alternating case, with an alternating basic entry sequence as in the preceding construction, and let $\mathcal{R}_R(I)$ denote its Rees algeb…

Let $R=k[x,y,z]$ be the polynomial ring in three variables over a field $k$, and let $P$ be the prime ideal defining the space monomial curve $(t^a,t^b,t^c)$ in $\mathbb A^3_k$. Fo…

Let $I$ be a monomial ideal of $k[x_1,\ldots,x_n]$ that is an almost complete intersection of finite colength, and let $\mathbf R=k[x_1,\ldots,x_n]$. Its Rees algebra is…

Let $A$ be the two-dimensional normal local ring under consideration, let $I\subset A$ be a $p_g$-ideal, and let $\mathcal{R}(I)=\bigoplus_{n\geq 0}I^nt^n$ denote its Rees algebra…

Let $k[\mathbf{x}]=k[x_1,x_2,x_3]$ and $k[\mathbf{y}]=k[y_1,y_2,y_3,y_4]$. Consider the matrix $\varphi:=\varphi_{3,r}$ as in the paper's template, and let…

Recursive elimination conjecture. For arbitrary $n$,

Monomial Rees-algebra conjecture.

Let $I$ be a monomial ideal of $k[x_1,\ldots,x_n]$. Suppose that $I$ is an almost complete intersection of finite colength, meaning that it has finite colength and one more minimal…