The end curve theorem for normal complex surface singularities

Walter D. Neumann; Jonathan Wahl

Displaying similar documents to “The end curve theorem for normal complex surface singularities”

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

Similarity:

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

A class of non-rational surface singularities with bijective Nash map

Camille Plénat, Patrick Popescu-Pampu (2006)

Bulletin de la Société Mathématique de France

Similarity:

Let $(𝒮, 0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_{i}$ , $i \in I$ . The Nash map associates to each irreducible component $C_{k}$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_{k}$ . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E \cdot E_{i} < 0$ for any $i \in I$ . ...

Quantum Singularity Theory for $A_{(r - 1)}$ and $r$ -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

Similarity:

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the $r$ -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity $W$ of type $A$ our construction of the stack of $W$ -curves is canonically isomorphic to the stack of $r$ -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an $r$ -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine...

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves such that $W (ℝ)$ is orientable. J. Kollár proved that a connected component $N$ of $W (ℝ)$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$ , our...

Springer fiber components in the two columns case for types $A$ and $D$ are normal

Nicolas Perrin, Evgeny Smirnov (2012)

Bulletin de la Société Mathématique de France

Similarity:

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element $N$ with $N^{2} = 0$ in a Lie algebra of type $A$ or $D$ (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let $C$ be a smooth curve of genus $g$ . For each positive integer $r$ the birational $r$ -gonality $s_{r} (C)$ of $C$ is the minimal integer $t$ such that there is $L \in {Pic}^{t} (C)$ with $h^{0} (C, L) = r + 1$ . Fix an integer $r \geq 3$ . In this paper we prove the existence of an integer $g_{r}$ such that for every integer $g \geq g_{r}$ there is a smooth curve $C$ of genus $g$ with $s_{r + 1} (C) / (r + 1) > s_{r} (C) / r$ , i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.

Singularities of $2 Θ$ -divisors in the jacobian

Christian Pauly, Emma Previato (2001)

Bulletin de la Société Mathématique de France

Similarity:

We consider the linear system $| 2 Θ_{0} |$ of second order theta functions over the Jacobian $J C$ of a non-hyperelliptic curve $C$ . A result by J.Fay says that a divisor $D \in | 2 Θ_{0} |$ contains the origin $𝒪 \in J C$ with multiplicity $4$ if and only if $D$ contains the surface $C - C = {𝒪 (p - q) ∣ p, q \in C} \subset J C$ . In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$ , divisors containing the fourfold $C_{2} - C_{2} = {𝒪 (p + q - r - s) ∣ p, q, r, s \in C}$ , and divisors singular along $C - C$ , using...

Foliations by curves with curves as singularities

M. Corrêa Jr, A. Fernández-Pérez, G. Nonato Costa, R. Vidal Martins (2014)

Annales de l’institut Fourier

Similarity:

Let $ℱ$ be a holomorphic one-dimensional foliation on $ℙ^{n}$ such that the components of its singular locus $Σ$ are curves $C_{i}$ and points $p_{j}$ . We determine the number of $p_{j}$ , counted with multiplicities, in terms of invariants of $ℱ$ and $C_{i}$ , assuming that $ℱ$ is special along the $C_{i}$ . Allowing just one nonzero dimensional component on $Σ$ , we also prove results on when the foliation happens to be determined by its singular locus.

A curve of genus $q$ with a Half-Canonical embedding in $\mathbf{P}^{3}$

Sevin Recillas (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Si costruiscono curve di genere $g=4n—3$ , $n\geq 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^{0}(L)=4$ . Per $n+3$ si dimostra che gli $L$ sono molto ampi.

On the complexity of braids

Ivan Dynnikov, Bert Wiest (2007)

Journal of the European Mathematical Society

Similarity:

We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $_{i j}$ , which are Garside-like half-twists involving strings $i$ through $j$ , and by counting powered generators $Δ_{i j}^{k}$ as $log (| k | + 1)$ instead of simply $| k |$ . The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism....

A local-global principle for rational isogenies of prime degree

Andrew V. Sutherland (2012)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let $K$ be a number field. We consider a local-global principle for elliptic curves $E / K$ that admit (or do not admit) a rational isogeny of prime degree $ℓ$ . For suitable $K$ (including $K = ℚ$ ), we prove that this principle holds for all $ℓ \equiv 1 mod 4$ , and for $ℓ < 7$ , but find a counterexample when $ℓ = 7$ for an elliptic curve with $j$ -invariant $2268945 / 128$ . For $K = ℚ$ we show that, up to isomorphism, this is the only counterexample.

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

Similarity:

We take up the study of the Brill-Noether loci $W^{r} (L, X) : = {η \in {Pic}^{0} (X) | h^{0} (L \otimes η) \geq r + 1}$ , where $X$ is a smooth projective variety of dimension $> 1$ , $L \in Pic (X)$ , and $r \geq 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^{0} (K_{D})$ , where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $> 2$ . In the $2$ -dimensional case...