36 problems

Ma's multiplicity conjecture.

Nonexistence conjecture. There is no Buchsbaum homogeneous integral domain over $k$ with minimal multiplicity of degree $q \ge 2$.

Huneke–Srinivasan–Herzog–Srinivasan conjecture. If $R/I$ is Cohen–Macaulay, then

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, and let $N$ be a finitely generated graded torsion $R$-module. Set $c=n-\dim N$ and, for a minimal grade…

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, let $I\subset R$ be a graded ideal, and set $c=\operatorname{codim} R/I$. For a minimal graded free reso…

Huneke–Srinivasan multiplicity conjecture. One has

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring with its standard grading, let $I\subset S$ be a graded ideal, and let $R=S/I$ be a standard graded $K$-algebra. Write…

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$,…

Let $x\in X$ be a point on a variety with log canonical singularities. Denote $e:=\dim m_x/m_x^2$ and $d:=\dim X$ at $x$. The minimal log discrepancy at $x$ is denoted by…

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ such tha…

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs, and let $f_*$ and $g_*$ be their initial forms. Suppose there is a homeomorphism…

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be holomorphic function-germs with isolated singularities at the origin. Suppose there is a homeomorphism…

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs, and let $V(f)$ and $V(g)$ denote their zero-set germs at the origin. Suppose there is a home…

Let $\mathcal{A}$ be a $k$-th order, $n$-dimensional tensor, let $\lambda$ be an eigenvalue, and let … be its eigenvariety. Suppose that this variety has irreducible components…

Let $X\subset \mathbb{C}^n$ and $Y\subset \mathbb{C}^m$ be complex analytic sets with $\dim X=\dim Y=d$. Their germs at zero are bi-Lipschitz homeomorphic if there is a bi-Lipschit…

Let $d\geq 2$ and $N\geq d$ be positive integers. Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d$, and let $I$ be an $\operatorname{\mathfrak{m}}$…

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring of dimension $d\geq 1$. Write $\widehat{R}$ for its completion, $\operatorname{Spec}\widehat{R}$ for its spectrum,…

Let $X,Y\subset\mathbb{R}^n$ be two germs at the origin of irreducible real analytic subsets. Let $h\colon(\mathbb{R}^n,0)\to(\mathbb{R}^n,0)$ be a germ of a subanalytic, arc-analy…

S-shape conjecture. The interval $(\underline{\lambda},\overline{\lambda})$ should be expandable so that problem … should be removable.

Let $X$ be an $n$-dimensional variety that is locally a complete intersection with log canonical singularities. For a closed point $x$ of $X$, let $\mathrm{emb}(X,x)$ be the embedd…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. Let $e(I,M)$ be the Hilbert–Samuel multiplicity of $M$ with respect to $I$, and se…

Let $(R,{\mathfrak m})$ be a regular local ring of dimension $d$, and let $t_i$ be defined by … An ideal is ${\mathfrak m}$-full if it has the corresponding maximal-ideal fullness…

Let $(R,{\mathfrak m})$ be a Noetherian local ring of dimension $d\geq4$, and let $\lambda(R/I)$ denote the length of $R/I$. The product-form generalized Lech inequality. For every…