Minimal rational graphs admitting a QHD smoothing

TL;DR

The paper advances the understanding of which complex surface singularities admit a $\mathbb{QHD}$ smoothing by combining De Jong–Van Straten's picture-deformation theory with a broad reduction framework. It proves nonexistence results for graphs with a small number of large nodes (or degree-3 nodes) and develops a finite-computation strategy via a reduction algorithm, coupled with a lattice-embedding constraint $Z_K^2+|\Gamma|=0$, to rule out broad families. A constructive portion identifies and analyzes numerous combinatorial smoothings (e.g., $fpp(n)$, $Cl(k,n)$, and $t(a,b,c)$-type configurations) and yields a new proof of the Bhupal–Stipsicz classification for weighted homogeneous singularities. The results connect the algebraic-deformation picture to topological implications (rational blowdowns) and extend Wahl’s conjecture program by clarifying which minimal rational and star-shaped graphs can admit $\mathbb{QHD}$ smoothings.

Abstract

Using the picture deformation technique of De Jong-Van Straten we show that no singularity whose resolution graph has 3 or 4 large nodes, i.e., nodes satisfying d(v)+e(v)\leq -2, has a QHD smoothing. This is achieved by providing a general reduction algorithm for graphs with QHD smoothings, and enumeration. New examples and families are presented, which admit a combinatorial QHD smoothing, i.e. the incidence relations for a sandwich presentation can be satisfied. We also give a new proof of the Bhupal-Stipsicz theorem on the classification of weighted homogeneous singularities admitting QHD smoothings with this method by using cusp singularities.

Figures (16)

• Figure 1: The graphs of $\mathcal{W},\mathcal{N}$ and $\mathcal{M}$ where $p,q,r\geq -1$.
• Figure 2: The starting graph for the construction of the $\mathcal{A},\mathcal{B},\mathcal{C}$ families. $(a,b,c)=(3,3,3),(2,4,4),(2,3,6)$ respectively.
• Figure 3: $f$ denotes the incidence matrix of a combinatorial smoothing, and $F_f$ the Milnor fiber (de1998deformation)
• Figure 4: The graph $Cl(k,n)$ for $k>1$. The number $x$ on an edge represents a path of $x$ many $-2$ vertices.
• Figure 5: The graph we obtain after cluster extending $Cl(k,n)$ for $k>1$. For $k<-1$, we still get some $\frac{p^2}{pq-1}$ linear graphs, which admit a $\mathbb{Q}\text{HD}$ smoothing.

Theorems & Definitions (97)

• Theorem 1.1
• Theorem 1.2: Bhupal-Stipsicz 2010 bhupal2011weighted
• Definition 2.1
• Theorem 2.2: stipsicz2008rational
• Theorem 2.3: stipsicz2008rational
• Theorem 2.4: stipsicz2008rational
• Definition 2.5
• Definition 2.6: de1998deformation
• Definition 2.7: de1998deformation
• Remark 2.8