In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allows us to recover some particular cases found in the literature.

We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$: in terms of the atypical fibres of $f$, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$.

We show that all large enough positive integral surgeries on algebraic knots bound a 4-manifold with a negative definite plumbing tree, which we describe explicitly. Then we apply the lattice embedding obstruction coming from Donaldson’s Theorem to classify the ones of the form $S^3_n(T(p_1,k_1p_1+1; p_2, k_2p_2\pm 1))$ that also bound rational homology 4-balls.

Let $(S,L)$ be a general polarised Enriques surface, with L not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible nodal curves in the linear system $|L|$ , that is, for any number of nodes $\delta =0, \ldots , p_a(L)-1$ . This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande–Schmitt, under the additional condition of non-2-divisibility.

Let k be a field of characteristic zero and let $\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if $\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing that the module ${\rm Hom}_A(\Omega_{A/k}, \Omega_{A/k})$ is then isomorphic to an ideal of A, say $\mathfrak{h}$, we show that A is regular whenever the ring $A/a\mathfrak{h}$ is Gorenstein for some parameter a (and conversely). In addition, we provide various characterizations for the regularity of A in the context of the conjecture.

Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\rightarrow S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D\subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected, but it can be made to vanish by a faithfully flat base change, if $S$ is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

We study the dynamics of a singular holomorphic vector field at $(\mathbb{C}^{2},0)$. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.

The motivic Hilbert zeta function of a variety $X$ is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points on $X$. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy $K$-theory groups of a Noetherian scheme $X$ of Krull dimension $d$ vanish below $-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy $K$-theory groups vanish below $-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy $K$-theory group.

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande–Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar–Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the ‘U(∞)’ invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

We prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for .

Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

The purpose of this paper is to develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniversal deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial (respectively, weakly equisingular) deformations, that is, deformations which become trivial (respectively, equisingular) after a finite and dominant base change. The semiuniversal base space for weakly equisingular deformations is, in general, not smooth but becomes smooth after a finite and purely inseparable base extension. The proof of this fact is more complicated and we introduce new constructions which may have further applications in the theory of singularities in positive characteristic.

We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact auto-equivalences and the set of all $t$-structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder–Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.

Patchworking of singular hypersurfaces is used to construct projective hypersurfaces with prescribed singularities. For all $n\,{\geq}\, 2$, an asymptotically proper existence result is deduced for hypersurfaces in $\P^n$ with singularities of corank at most 2 prescribed up to analytical or topological equivalence. In the case of $T$-smooth hypersurfaces with only simple singularities, the result is even asymptotically optimal, that is, the leading coefficient in the sufficient existence condition cannot be improved, which is new even in the case of plane curves. Furthermore, an asymptotically proper existence result is proved for hypersurfaces in $\P^n$ with quasihomogeneous singularities. The estimates substantially improve all known (general) existence results for hypersurfaces with these singularities.

Bertini's theorem on variable singular points may fail in characteristic $p$; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations $ zx^3 + xy^3 + t yz^3 = 0$.

More generally, we classify pathological fibrations by canonical curves of arithmetic genus $g = \frac{p - 1}{2}$ in the projective space of dimension $\frac{p - 3}{2}$. We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer $\ell$ is a gap if and only if $2g + 1 - \ell$ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces.