286 problems

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 $k$ be a field of characteristic $0$, let $n\ge 1$, and let $f\in (x_1,\ldots,x_n)^2\subset k[[x_1,\ldots,x_n]]$ define an isolated critical point at the origin. If…

For every holomorphic map germ $\boldsymbol{n}:(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ that admits an injective representative, the differential at the origin is nonzero:…

For every nonzero ideal $I\subseteq\mathbb{C}[x_1,\ldots,x_n]$, every pole $s_0$ of a local topological zeta function associated with $I$ is a root of the Bernstein–Sato polynomial…

Let $(R,\mathfrak m,k)$ be an $F$-finite, unmixed, $d$-dimensional Noetherian local ring of characteristic $p>0$. If $R$ is not regular, then its Hilbert–Kunz multiplicity satisfie…

Let $f$ and $g$ be isolated hypersurface singularities, and let $f\oplus g=f(x)+g(y)$ be their separated Thom–Sebastiani sum. Denote by $\operatorname{TjSp}(f)$ the Tjurina subspec…

Every reduced irreducible complex plane curve $C\subset\mathbb{P}^{2}$ whose normalization has genus zero and whose singularities are all unibranch is either free or nearly free.

Let $X$ be a smooth complex affine variety of dimension $n$, let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions such that $f=\prod_{i=1}^r f_i$ is not invertible, and…

For a smooth threefold $X$ and an integer $\ell\geq 1$, characterize the points $[Z]\in \operatorname{Hilb}^{\ell}(X)$ having maximal singularity, namely those satisfying…

Let $\mathfrak a\subseteq \mathbb C[x_1,\ldots,x_m]$ and $\mathfrak b\subseteq \mathbb C[y_1,\ldots,y_n]$ be nonzero ideals, and regard them as ideals in…

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$, with $n\ge 2$ and $\operatorname{mult}(f)\ge 3$. Let $A(V)$ be the moduli…

For every tuple $F=(f_1,\ldots,f_r)$ of nonzero holomorphic germs on a smooth complex germ $(X,0)$, every actual polar hyperplane $H$ of the local multivariable topological zeta fu…

Machado–Seade conjecture. The following statements are equivalent: $(D,0)$ is weighted homogeneous; there exists a holomorphic vector field tangent to $D$ with an isolated singular…

Hertling's conjecture. The spectrum numbers satisfy

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…

Strong Lefschetz conjecture. For any smooth $F\in S_d$, any integer $k\in[0,T/2)$, and any generic linear form $\ell\in S_1$, the induced multiplication map is an isomorphism. The…

Let $f$ be an isolated quasihomogeneous singularity. Let $(M_1,\ldots,M_{l_\psi})$ be the standard covering associated with its weight-system data, and let $(H_{Mil},h_{Mil})$ deno…

Yano's conjecture. For generic curves in some $\Gamma$-constant deformation of $f$, the $b$-exponents are

Let $f$ be a non-constant polynomial in $\mathbb{Z}[x_1,\ldots,x_n]$, for some $n\geq 1$, with $f(\mathbf{0})=0$. For every prime number $p$, let $Z_{f,p}(s)$ be Igusa's $p$-adic z…

Equivalence conjecture. The following statements are equivalent:

Let $f \in \mathbb{C}[x_1,\dots,x_n] \setminus \mathbb{C}$ and assume $f(o)=0$. The local motivic zeta function $Z_{\operatorname{mot},o}(f;s)$ and the local topological zeta funct…

Let $f\colon (\mathbb{C}^n,S) \to (\mathbb{C}^{n+1},0)$ be an mathcal{F}-finite frontal map germ. Let $\mu_{\mathcal{F}}(f)$ denote its frontal Milnor number, and let…

Let $X$ be a $\mathbb Q$-Fano threefold, and let $H\in|-K_X|$ be a generic section of the anticanonical divisor. Reid's general elephant conjecture. The surface $H$ has at worst ca…

Let $f\in\mathbb{C}[[x,y,z]]$ be an isolated hypersurface singularity. Let $\mu$ be its Milnor number and $p_g$ its geometric genus. Durfee's conjecture. … with equality only when…

Shifted-binomial log-concavity conjecture. The shifted polynomial