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Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces

Ishan Banerjee, Nick Salter

TL;DR

<3-5 sentence high-level summary>: The paper determines the topological monodromy of sufficiently ample linear systems on smooth simply connected surfaces, proving it equals the r-spin mapping class group associated to the maximal root of the adjoint line bundle. It develops the theory of r-spin mapping class groups, framings, and wind­ing-number functions, and uses assemblage-based generation to produce a comprehensive set of vanishing cycles. By decomposing line bundles and employing smoothing techniques, it constructs a rich core and leverages braid-group methods and plane-curve singularity monodromies to realize the full r-spin monodromy. The main result yields a precise vanishing-cycle description and has corollaries for discriminants, Lefschetz fibrations, and 4-manifold embeddings, with broad implications for symplectic and algebraic geometry.

Abstract

We compute the mapping class group-valued monodromy of any sufficiently ample linear system on any smooth simply connected projective surface, identifying this with the r-spin mapping class group associated to a maximal root of the adjoint line bundle. This gives a characterization of the simple closed curves that can arise as vanishing cycles for nodal degenerations in the linear system, as well as other corollaries concerning discriminants, Lefschetz fibrations, and surfaces in 4-manifolds.

Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces

TL;DR

<3-5 sentence high-level summary>: The paper determines the topological monodromy of sufficiently ample linear systems on smooth simply connected surfaces, proving it equals the r-spin mapping class group associated to the maximal root of the adjoint line bundle. It develops the theory of r-spin mapping class groups, framings, and wind­ing-number functions, and uses assemblage-based generation to produce a comprehensive set of vanishing cycles. By decomposing line bundles and employing smoothing techniques, it constructs a rich core and leverages braid-group methods and plane-curve singularity monodromies to realize the full r-spin monodromy. The main result yields a precise vanishing-cycle description and has corollaries for discriminants, Lefschetz fibrations, and 4-manifold embeddings, with broad implications for symplectic and algebraic geometry.

Abstract

We compute the mapping class group-valued monodromy of any sufficiently ample linear system on any smooth simply connected projective surface, identifying this with the r-spin mapping class group associated to a maximal root of the adjoint line bundle. This gives a characterization of the simple closed curves that can arise as vanishing cycles for nodal degenerations in the linear system, as well as other corollaries concerning discriminants, Lefschetz fibrations, and surfaces in 4-manifolds.

Paper Structure

This paper contains 32 sections, 51 theorems, 112 equations, 11 figures.

Table of Contents

  1. Introduction
  2. $r$-spin mapping class groups
  3. Framings and $r$-spin structures
  4. $r$-spin mapping class groups and their generating sets
  5. Functoriality
  6. Non-containment implies maximality
  7. Sections of (sums of) line bundles
  8. The monodromy groupoid
  9. Topological monodromy of a linear system and $r$-spin structures
  10. Nodal degenerations, vanishing cycles and Picard-Lefschetz theory
  11. The topological model: $\widetilde{C}, \widetilde{D}$, and $E$
  12. The (principal) stabilizer
  13. Loci of smoothings
  14. Changing basepoints
  15. (Simple) braid groups
  16. ...and 17 more sections

Key Result

Theorem 1

Let $X$ be a smooth projective surface with $\pi_1(X) = 1$. Let $L$ be a line bundle on $X$ expressible as $L = L_1 \otimes L_2$, where Let $E \in \left| L \right|$ be a general smooth curve, and let $\Gamma_L \leqslant \mathop{\mathrm{Mod}}\nolimits(E)$ denote the topological monodromy group. Let $\phi_M$ be the $r$-spin structure associated to the maximal $r^{th}$ root $M$ of the adjoint line b

Figures (11)

  • Figure 1: Smoothing transversely-intersecting $C \in U_{L_1}$ and $D \in U_{L_2}$ (shown in color versions in blue and red, respectively) to a section $E = \widetilde{C} \cup \widetilde{D}$ of $\left| L_1 \otimes L_2 \right|$.
  • Figure 2: At left: two basic examples of surface braids: a simple half-twist about an arc, and a point-push. At right, an illustration of the "follow to the right" convention used in defining simple half-twists, and the simple closed curve $\sigma^+$ associated to an arc $\sigma$.
  • Figure 3: Three boundary-adjacent curves/spheres, one of which is maximal.
  • Figure 4: The subsurface $S_0 \subset E$ and the vanishing cycles $a_1, \dots, a_7, b_1, \dots, b_6$.
  • Figure 5: At left, the five-holed boundary-adjacent sphere $B'$, as embedded in $S_0\subset E$, with $S_0 \subset E$ a subsurface as in \ref{['fig:core']}. At right, an abstract view of a five-holed sphere depicting the curves $x,y,z,w$.
  • ...and 6 more figures

Theorems & Definitions (117)

  • Theorem 1
  • Remark 1.1: Jet ampleness
  • Remark 1.2: The ampleness hypothesis
  • Remark 1.3: Why simple connectivity?
  • Remark 1.4: $r$ = 1?
  • Remark 1.5: "Curves" and "surfaces"
  • Corollary 1.6
  • proof
  • Corollary 1.7
  • Corollary 1.8
  • ...and 107 more