Pluricanonical cycles and tropical covers

TL;DR

This work extends double Hurwitz theory to the pluricanonical setting by introducing logarithmic double ramification cycles $ extsf{DR}_g^{ m log}(oldsymbol{x},k)$ and a corresponding tropical framework. A central construction is the branch polynomial br_g(oldsymbol{x},k), with intersection numbers $L_g(oldsymbol{x},k)= ext{deg}([ extsf{DR}_g^{ m log}(oldsymbol{x},k)]igr floor br_g(oldsymbol{x},k))$ that are computable via tropical leaky covers and admit a piecewise-polynomial dependence on the ramification data. The authors connect these counts to a bosonic Fock-space operator $M_k$, showing that $L_g(oldsymbol{x},k)$ are matrix elements $iglraket{b_{oldsymbol{x}^+}ig| M_k^{n-2+2g}ig| b_{-oldsymbol{x}^-}igr}$ up to standard symmetry factors, and they recover lower-genus Hurwitz data by level-h branch polynomials br_h<g. The paper thus provides a coherent tropical-analytic pipeline from logarithmic intersection theory to explicit enumerative formulas, offering a versatile toolkit for pluricanonical intersection theory and potential applications in logarithmic Gromov–Witten theory.

Abstract

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

Figures (14)

• Figure 1: A $1$-leaky cover of genus $1$ and degree $(5,-1,-1)$, with its minimal vertex set. We do not specify length data in this picture, as the lengths in $\Gamma$ are imposed by the distances of the points in $T$. All vertices are supposed to be of genus $0$. For simplicity, we also suppress the labels for the ends in this picture.
• Figure 2: The tautological diagram used to express $H_g(\mathbf{x})$ as an intersection number on $\overline{{\mathcal{M}}}_{g,n}$.
• Figure 3: The subdivision of the moduli space $\mathcal{M}^{\mathrm{trop}}_{1,2}$ induced by the moduli space of tropical rubber maps in genus $1$ with contact $(3,-3)$.
• Figure 4: The birational transformation of $\overline{\mathcal{M}}_{1,2}$ for the $(3,-3)$ double ramifucation problem in genus $1$ and the dimensionally transverse cycle $\mathsf{DR}_{g}^{\circ,{\mathsf{log}}}(\mathbf{x},0)$. We denote by $\mathsf{DR}_1^{\circ} ((3,-3),0)$ the closure of the component of the space of relative stable maps where for the general point the source curve is smooth.
• Figure 5: A tropical cover in $M^{\mathrm{trop}}_{2<6}(4,1,-2,-3)$ corresponding to a maximal dimensional cone.

Theorems & Definitions (62)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.1.1
• Definition 2.2.1
• Definition 2.2.2: Leaky cover
• Remark 2.2.3: Vertex set
• Definition 2.2.4: Left and right degree
• Definition 2.2.5: Divisors and the tropical canonical
• Definition 2.2.6: Rational functions on abstract tropical curves and their divisors