arXiv subjects

Jonathan Wahl

Publications and source records attributed to Jonathan Wahl.

18 recordsLinked to original sources

arXiv2023

The Volume of a Surface or Orbifold Pair

A surface pair $(X,C)$ is a germ of a normal surface singularity $(X,0)$ and a sum $C=\sum c_iC_i$ of curves on $X$, with $c_i\in [0,1]$. An orbifold pair has $c_i=1/n_i$, as intersecting with a small sphere gives a $3$-dimensional orbifold $(\Sigma, \gamma_i,n_i)$. There are natural notions of morphism and log cover of surface pairs. We introduce a volume $Vol(X,C)$ in $\mathbb Q_{\geq 0}$, computable from any log resolution, analogous to that in our 1990 JAMS paper when $C=0$. Denoting $\bar{C}=\sum (1-c_i)C_i$, one has $(X,C)$ log canonical iff $Vol(X,\bar{C})=0$. The main theorem (5.4-5.6) is that $Vol(X,C)$ is ``characteristic'': if $f:(X',C')\rightarrow (X,C)$ is a morphism of degree $d$, then $Vol(X',C')\geq d\cdot Vol(X,C)$, with equality if $f$ is a log cover. We prove (6.7) that $Vol(X,\sum (1/n_i)C_i)=0$ iff the associated orbifold has finite or solvable orbifold fundamental group, and these are classified. In (8.2) is proved a key case of the DCC Volumes Conjecture: The set $\{Vol(X,\sum (1/n_i)C_i)|\ X\ \text{RDP}\}$ satisfies the DCC, with minimum non-$0$ volume 1/3528.

math.AG↗

arXiv2022

Rational homology disk smoothings of surface singularities; the exceptional cases

It is known (Stipsicz-Szabó-Wahl) that there are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a rational homology disk ($\mathbb{Q}$HD) smoothing, i.e., having a Milnor fibre with Milnor number zero. Some examples are found by an explicit "quotient construction", while others require the "Pinkham method". The fundamental group of the Milnor fibre has been known for all except the three exceptional families $\mathcal B_2^3(p), \mathcal C^3_2(p),$ and $\mathcal C^3_3(p)$. In this paper, we settle these cases. We present a new explicit construction for the $\mathcal B_2^3(p)$ family, showing the fundamental group is non-abelian (as occurred previously only for the $\mathcal A^4(p), \mathcal B^4(p)$ and $\mathcal C^4(p)$ cases). We show that the fundamental groups for $ \mathcal C^3_2(p)$ and $\mathcal C^3_3(p)$ are abelian, hence easily computed; using the Pinkham method here requires precise calculations for the fundamental group of the complement of a plane curve.

math.AG↗

arXiv2022

Splice diagrams and splice-quotient surface singularities

The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to define singularities of splice type and splice-quotient singularities. The Casson Invariant Conjecture and Milnor Fiber Conjectures are discussed, as well as some open questions about hypersurface and complete intersection singularities with integral homology sphere link.

math.AG↗

arXiv2021

Orbifold splice quotients and log covers of surface pairs

A three-dimensional orbifold $(Σ, γ_i, n_i)$, where $Σ$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where $C$ is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients.

math.AG↗

arXiv2020

Complex surface singularities with rational homology disk smoothings

A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szabó, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each case, except for types $\mathcal W$, $\mathcal N$, and $\mathcal M$. In this paper, we describe his results, and settle these remaining cases; there is a unique $\mathbb Q$HD smoothing component except in the cases of an obvious symmetry of the resolution dual graph. The method involves study of configurations of rational curves on projective rational surfaces.

math.AG↗

arXiv2016

The number of equisingular moduli of a rational surface singularity

We consider a conjectured topological inequality for the number of equisingular moduli of a rational surface singularity, and prove it in some natural special cases. When the resolution dual graph is "sufficiently negative" (in a precise sense), we verify the inequality via an easy cohomological vanishing theorem, which implies that this number is computed simply from the graph (Theorem 3.10). To consider an important and less restrictive meaning of "sufficiently negative" requires a much more difficult "hard vanishing theorem" (Theorem 4.5), which is false in characteristic p. Theorem 7.9 verifies the conjectured inequality in this more general situation. As a corollary, we classify in characteristic p all taut singularities with reduced fundamental cycle (Theorem 9.2).

math.AG↗

arXiv2016

Milnor and Tjurina numbers for smoothings of surface singularities

For an isolated hypersurface singularity $f=0$, the Milnor number $μ$ is greater than or equal to the Tjurina number $τ$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity $(V,0)$, the difference $μ- τ$ can be defined whether or not $V$ is smoothable; it was proved in [23] that it is non-negative, and equal to 0 iff $(V,0)$ is quasi-homogeneous. We conjecture a similar result for non-Gorenstein surface singularities. Here, $μ- τ$ must be modified so that it is independent of any smoothing. This expression, involving cohomology of exterior powers of the bundle of logarithmic derivations on the minimal good resolution, is conjecturally non-negative, and equal to 0 iff one has quasi-homogeneity. We prove the "if" part; identify special cases where the conjecture is particularly interesting; verify it in some non-trivial cases; and prove it for a $\Q$Gorenstein smoothing when the index one cover is a hypersurface. This conjecture is of interest regarding the classification of surface singularities with rational homology disk smoothings, as in [1], [18], [24].

math.AG↗

arXiv2013

Log-terminal smoothings of graded normal surface singularities

Recent work ([18], [1]) has produced a complete list of weighted homogeneous surface singularities admitting smoothings whose Milnor fibre has only trivial rational homology (a "rational homology disk"). Though these special singularities form an unfamiliar class and are rarely even log-canonical, we prove the Theorem. A rational homology disk smoothing of a weighted homogeneous surface singularity can always be chosen so that the total space is log-terminal. In particular, this smoothing is $\Q$-Gorenstein. The key idea is to define a finite "graded discrepancy" of a normal graded domain with $\Q$-Cartier canonical divisor, and to study its behavior for a smoothing.

math.AG↗

arXiv2011

On rational homology disk smoothings of valency 4 surface singularities

Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are the three classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes in a previous paper; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different QHD smoothings for the first class. We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H-shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal-Stipsicz, is evidence for a general Conjecture: The only complex surface singularities with a QHD smoothing are the (known) weighted homogeneous examples.

math.AG↗

arXiv2008

The End Curve Theorem for normal complex surface singularities

We prove the "End Curve Theorem," which states that a normal surface singularity $(X,o)$ with rational homology sphere link $Σ$ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An "end-curve function" is an analytic function $(X,o)\to (\C,0)$ whose zero set intersects $Σ$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A "splice-quotient singularity" $(X,o)$ is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in $\C^t$, where $t$ is the number of leaves in the resolution graph for $(X,o)$, together with an explicit description of the covering transformation group. Among the immediate consequences of the End Curve Theorem are the previously known results: $(X,o)$ is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).

math.AG↗

arXiv2008

Rational blow-downs and smoothings of surface singularities

In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are also presented; these include singularities with resolution graph having valency four nodes.

math.GT↗

arXiv2005

Topology, geometry, and equations of normal surface singularities

This expository talk is an expanded version of a lecture at G.-M. Greuel's 60th Birthday Conference in Kaiserslautern in October, 2004. We survey recent work of Neumann-Wahl and others on the relation between topology and geometry of normal surface singularities, especially how to write explicit equations of a singularity with given topology, in case the link is a rational homology sphere (i.e., the resolution graph is a tree of rational curves). Specifically, under some reasonable conditions on the link, there is an explicit "complete intersection of splice type" whose link is the universal abelian cover of the original one, and for which the action of the covering group can be explicitly described. Recent work of T. Okuma shows every rational surface singularity can be described in this way. Full details are found in papers in Geometry and Topology, math.AG/0301165 and math.AG/0407287.

math.AG↗

arXiv2005

Complete intersection singularities of splice type as universal abelian covers

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with Q-homology sphere links. As quotients of complete intersections, they are necessarily Q-Gorenstein, and many Q-Gorenstein singularities with Q-homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with Q-homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.

math.AG↗

arXiv2005

Complex surface singularities with integral homology sphere links

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].

math.AG↗

arXiv2001

Universal abelian covers of surface singularities

We discuss the evidence for and implications of a conjecture that the universal abelian cover of a Q-Gorenstein surface singularity with finite local homology (i.e., the singularity link is a Q-homology sphere) is a complete intersection singularity.

math.AG↗

arXiv2001

Hyperplane sections of Calabi-Yau varieties

Theorem: If W is a smooth complex projective variety with h^1 (O-script_W) = 0, then a sufficiently ample smooth divisor X on W cannot be a hyperplane section of a Calabi-Yau variety, unless W is itself a Calabi-Yau. Corollary: A smooth hypersurface of degree d in P^n (n >= 2) is a hyperplane section of a Calabi-Yau variety iff n+2 <= d <= 2n+2. The method is to construct out of the variety W a universal family of all varieties Z for which X is a hyperplane section with normal bundle K_X, and examine the "bad" singularities of such Z. A motivation is to show many curves cannot be divisors on a K-3 surface.

math.AG↗

arXiv2001

Universal abelian covers of quotient-cusps

The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a $\Q$-Gorenstein singularity.

math.AG↗

arXiv1996

On Cohomology of the Square of an Ideal Sheaf

For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$, $k\ge3$. This condition is shown to be satisfied for all sufficiently large embeddings of a given $X$, and for a Veronese embedding of $\Bbb P^n$. For $C\subset\Bbb P^{g-1}$, the canonical embedding of a non-hyperelliptic curve, this condition guarantees the vanishing of some obstruction groups to deformations of the cone. Recall that the tangents to deformations are dual to the cokernel of the Gaussian-Wahl map. \proclaim{Theorem} Suppose the Gaussian-Wahl map of $C$ is not surjective and the vanishing condition is fulfilled. Then $C$ is {\bf extendable}: it is a hyperplane section of a surface in $\Bbb P^g$ not the cone over $C$.\endproclaim Such a surface is a K3 if smooth, but it could have serious singularities. \proclaim{Theorem} For a general curve of genus $\ge3$, this vanishing holds. \endproclaim \proclaim{Conjecture} If the Clifford index is $\ge3$, this vanishing holds. \endproclaim

alg-geom↗