597 problems

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 $f:(\mathbb{C}^m,0)\to(\mathbb{C}^{m+1},0)$ be an $\mathcal{A}$-finite germ in the Mather range, meaning $m\leq M(1)=14$. Its image Milnor number is denoted by $\mu_I(f)$, and…

Let $p$ be a prime and let $f\in\mathbb{Z}[x_1,\dots,x_d]$. The poles of the Igusa zeta function encode the asymptotic behavior of the numbers of solutions of $f\equiv 0\pmod{p^k}$…

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 $F:(\mathbb{C}^{n+1},\mathbf{0})\to(\mathbb{C},\mathbf{0})$ define a germ of an isolated hypersurface singularity. For $x\in F^{-1}(0)$, let $\mathcal{F}_{F,x}$ be the Milnor f…

Hertling's conjecture. The spectrum numbers satisfy

Let $f:({\mathbb C}^{m+1},0)\to({\mathbb C},0)$ be an isolated quasihomogeneous hypersurface singularity, let $(H_{\mathbb Z},M)$ be its associated integral lattice with monodromy,…

Let $S_t\subset\mathbb{R}^3$ be a family of compact surfaces evolving by mean curvature flow, with generic initial data $S_0$. A singularity is spherical or cylindrical when its ta…

Let $X$ be a variety over a perfect field $k$. Write $\operatorname{Der}_{\mathbb{C}}(\mathcal{O}_X)$ for the module of derivations of $\mathcal{O}_X$ over $\mathbb{C}$. Zariski–Li…

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…

Let $F=(f_1,\ldots,f_r)$ be a tuple of polynomials, let $Z_F^{mot}(s_1,\ldots,s_r)$ be its multivariable motivic zeta function, and let $B_F\subset\mathbb{C}[s_1,\ldots,s_r]$ be it…

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

Let $f:{\mathbb{C}}^n\to {\mathbb{C}}^m$ be a map, and let its Thom polynomials be expressed in the Chern classes of $f$. Rimányi's Chern positivity conjecture. These Thom polynomi…

Let $f:\mathbb{R}^4\to\mathbb{R}^2$ be a polynomial map with an isolated singularity at the origin, meaning that $f(0)=(0,0)$, its first derivatives vanish at the origin, and its J…

Let $f$ be a polynomial, and let the complexification of $f$ have monodromy eigenvalues. Consider the Igusa zeta function of $f$ with respect to a character, and let the order of t…

Let a family of singularities have constant Milnor number, denoted by $\mu$; Whitney conditions are the usual geometric regularity conditions for the family. Teissier's conjecture.…

Let $X_t$ be a family of hypersurfaces in $\mathbb{C}^3$ parameterized by finitely determined map germs $f_t$, and let $D(f_t)$ denote the double point curve of $f_t$. Ruas' conjec…

Let $f\colon (\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ be an injective holomorphic map germ. A map germ is corank $1$ when the differential $df_0$ has corank $1$. Lê's conjecture.…

Aluffi–Harris conjecture. The same formula should admit a natural generalization to arbitrary projective varieties, including singular ones, by replacing the Euler characteristic w…

$X_9^2$ component conjecture. These representatives exhaust the components of the complement of the discriminant variety; in total there are 52 such components.

Let $f\in\mathbb{C}[[x,y]]$ be an isolated hypersurface singularity, with Milnor number $\mu$ and Tjurina number $\tau$. Dimca–Greuel's conjecture. … The conjecture was proved comp…

Let stable maps be maps whose qualitative behavior is unchanged under the relevant coordinate changes, and allow all homeomorphisms as coordinate changes. Thom's density conjecture…

Let $Q$ be a regular local ring of dimension $n+1$ with algebraically closed residue field, and let $e$ denote the floor of $\frac{n}{2}$, which we write as…

Maximal Tjurina existence conjecture. For every integer $d\geq 3$ and every integer $r$ with $d/2\leq r\leq d-1$, there are maximal Tjurina curves of type $(d,r)$. Moreover, for…

Let $\mathbf{w}$ be an invertible polynomial in $n$ variables, let $\Gamma_{\mathbf{w}}$ be its maximal grading group, and let $\widecheck{\mathbf{w}}$ be its Berglund–Hübsch trans…