The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

We study pencils of curves on a germ of complex reduced surface $(S,0)$. These are families of curves parametrized by $ \mathbb{P}^1 $ having 0 as the unique common point. We prove that for $w\in \mathbb{P}^1$, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$, where f and g are generators of the pencil.

Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{\mathrm {reg}}$, and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.

We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${\Bbb R}$-ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.

We characterize the finite codimension sub-${\mathbf {k}}$-algebras of ${\mathbf {k}}[\![t]\!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${\mathbf {k}}$-vector spaces of ${\mathbf {k}}[u]$, this ring acts on ${\mathbf {k}}[\![t]\!]$ by differentiation.

Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$. We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$-invariant, but lie in distinct path-connected components of the $\mu ^*$-constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$, respectively) make a Zariski pair of curves in $\mathbb {P}^2$, the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$-Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.

We consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by $\gamma_f$, in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ $g\,{:}\,(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $\gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.

Given a resolution of rational singularities $\pi \colon {\tilde {X}} \to X$ over a field of characteristic zero, we use a Hodge-theoretic argument to prove that the image of the functor ${\mathbf {R}}\pi _*\colon {\mathbf {D}}^{\mathrm {b}}({\tilde {X}}) \to {\mathbf {D}}^{\mathrm {b}}(X)$ between bounded derived categories of coherent sheaves generates ${\mathbf {D}}^{\mathrm {b}}(X)$ as a triangulated category. This gives a weak version of the Bondal–Orlov localization conjecture [BO02], answering a question from [PS21]. The same result is established more generally for proper (not necessarily birational) morphisms $\pi \colon {\tilde {X}} \to X$, with ${\tilde {X}}$ smooth, satisfying ${\mathbf {R}}\pi _*({\mathcal {O}}_{\tilde {X}}) = {\mathcal {O}}_X$.

Let f be an isolated singularity at the origin of $\mathbb {C}^n$. One of many invariants that can be associated with f is its Łojasiewicz exponent $\mathcal {L}_0 (f)$, which measures, to some extent, the topology of f. We give, for generic surface singularities f, an effective formula for $\mathcal {L}_0 (f)$ in terms of the Newton polyhedron of f. This is a realization of one of Arnold’s postulates.

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$ -Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann. 271 (1985), 439–449] and S. Ishii [Math. Ann. 275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].

Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Kollár component of $X\ni x$ . If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$ . This extends a result of Blum and Kawakita who independently showed that any divisor computing the minimal log discrepancy on a smooth surface is a potential lc place.

Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.

Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.

Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity.

Utilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $\mathcal {B}(R)$ over a local domain R essentially of finite type over $\mathbb {C}$. We show that if R is normal and $\Delta $ is an effective $\mathbb {Q}$-Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$-Cartier, then the BCM test ideal $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ coincides with the multiplier ideal $\mathcal {J}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$, where $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$-adic completions of R and $\mathcal {B}(R)$, respectively, and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$. As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.

It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots , f^{k_0}$ if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ is a nondegenerate complete intersection variety for any $k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$ .

In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.

We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper, C. Bonnafé [‘Calogero–Moser spaces vs unipotent representations’, Pure Appl. Math. Q., to appear, Preprint, 2021, arXiv:2112.13684].

We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.