34 problems

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…

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 $A$ be a polynomial ring over a field $k$. Let ${\mathfrak m}$ be the maximal graded ideal of $A$, and let $I$ be an equigenerated homogeneous ${\mathfrak m}$-primary ideal. Wr…

Vasconcelos's conjecture. The Rees algebra $\mathcal{R}(I)$ is almost Cohen--Macaulay, meaning

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…

Conca–Herzog–Trung–Valla conjecture. If $R_A(I)$ is a Cohen–Macaulay ring, then there exists a diagonal $\Delta$ such that $R_A(I)_\Delta$ is a Cohen–Macaulay ring.

Let $\mathbb X$ be an arbitrary set of points in $\mathbb P^2$, and let … be its defining ideal. Conjecture. For every non-negative integer $p$, there exists an integer $d_p$ such…

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

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

Fiber-type classification conjecture. (1) If

Let $(B,\mathfrak{n})$ be a complete local ring containing a field of characteristic $p>0$, and let $I$ be a proper ideal in $B$. Write $G(I)=\bigoplus_{n\geq 0}I^n/I^{n+1}$ for th…

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 $I_n(B)$ 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 $I$ be the ideal of a linear star configuration in a polynomial ring, obtained by choosing arbitrary hyperplanes in projective space. An ideal $I$ is of fiber type if the non-l…

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 $I_{n,h}$ denote the family of ideals in Proposition, and let $\operatorname{svd}(I_{n,h})$ be its standard Veronese degree. Symbolic Veronese degree conjecture. We conjecture…

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 $I$ 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 $(R,\mathfrak m,\Bbbk)$ be a three-dimensional Cohen–Macaulay analytically unramified local ring, let $I$ be an $\mathfrak m$-primary ideal, and let … be the extended Rees ring…