Fixed Tropical Type To The Base Residue -- Algorithm For The Artin-Base Term In Relative Virtual Localization

algorithm

The point of this note is to give an algorithm for taking a fixed tropical type and producing the base residue. Work in progress. We’ll fix \(\mathbb P^2\) with full toric log structure as our target for now.

Inputs

The following are inputs, they are subject to certain conditions. For instance demanding that a type be fixed incurs conditions on the type equivalent to the Kontesevich trees from ordinary GW theory and the balancing condition at the vertices of the dual graph \(G\) often completely determines the contact orders of the edges from the contact orders of the legs. Geometric

  • \(X = \mathbb P^2\), \(T = \mathbb G_m^2\),
  • \(M = \mathbb Zm_1\oplus \mathbb Zm_2 = \Hom(T, \mathbb G_m)\)
  • \(N = \mathbb Z\rho_1 \oplus \mathbb Z\rho_2\) with \(\rho_0 = -\rho_1-\rho_1\), then the fan of \(\mathbb P^2\) is the complete fan with rays \(\rho_0,\rho_1,\rho_2\). Choose \(\rho_1 = m_1^\vee\) and \(\rho_2 = m_2^\vee\).
  • write \(D_i = \{x_i = 0\}\) for the toric divisors. We write \(x_1/x_0 = x^{m_1}\) and \(x_2/x_0 = x^{m_2} \).
  • write \(p_{ij} = D_i\cap D_j\) for the three fixed points of \(X\).
  • write \(\lambda(p, D)\in M\) for the tangent weight of \(T\) at \(p\) along the divisor \(D\).
  • the moduli space \(\mathcal M = \mathcal M(X)\) of basic stable log maps to \(X\)
  • a curve class \((C, f, y_1,...,y_n)\in \mathcal M\)

We’ll later write \(\lambda_v(x) \in \frac{1}{d_v}M\) for the tangent weight of the cover \(C_v\) at a special point \(x\) over \(p\) when \(f|_{C_v}:C_v\to D\) is a degree \(d_v\) cover.

Discrete

  • the type \(\tau = (G, \mathbf g, \boldsymbol \sigma, \{u_y, u_q\})\)
    • a graph \(G\) comprised of legs \(L(G)\), edges \(E(G)\) and vertices \(V(G)\)
    • genus map \(\mathbf g:V(G)\to \mathbb N\)
    • cone map \(\boldsymbol \sigma:G\to \Sigma(X)\) assigning a cone to each part of the graph \(G\)
    • contact orders \(u_q\) and \(u_y\) at the nodes and marks
  • the basic monoid \(Q_\tau\), \(H\subset Q_\tau\) a Hilbert basis, \(V = \Spec k[Q_\tau]\), \(T_V = \Spec k[Q_\tau^{gp}]\), used in the local model of the algebraic stack base

Restrictions on the inputs

  • The curve \((C, f, z)\) is fixed by the \(T\) action and \(\tau\) is its type – in particular \(\tau\) is realizable.

Derived Quantities

D1: Structure of covers and geometric contact components

Lemma

If \(C_v\) is a non-conectracted component of \(C \) mapping to \(D_i\), then it is totally ramified of degree \(d_v\) and \(C_v\cong \mathbb P^1\). Its two ramification points \(x_\pm(v)\) lie over the two fixed points of \(D_i\) and are its only special points. The map \(f|_{C_v} \) is given in local coordinates by \(z\mapsto z^{d_v}\).

Proof

See this note.

Lemma

If \(x \in \{x_\pm(v)\}\) lies over \(p_{ij}\) and \(m\in \rho_i^\perp\cap M\) is the affine coordinate of \(D_i\) vanishing at \(p_{ij}\) (that is, \(x^m\in k[\sigma_{ij}^\vee \cap M]\)) is the local coordinate of \(D_i\) in an affine cover) then

\begin{align*} \langle u^{out}_x, m\rangle = d_v \end{align*}

where \(u^{out}_x\) is the contact order datum defined by a flag of the graph; if \(x\) is a marked point \(y\) then this is merely the contact order \(u_y\) but at an edge it flips sign depending on the choice of orientation.

D2: Balancing Condition

Lemma

At every vertex \(v\) of \(G\),

\begin{align*} \sum_{v\in x} u^{out}_x = 0 \end{align*}

where the sum is over all special points of the component \(C_v\). When \(G\) is a tree this determines all \(u_e\) from the \(u_z\).

Part 0: Fix the discrete data

Part 5: Isomorphism data of the fixed stable map is unique

Here’s a lemma about automorphisms of a log point, see also morphisms to and from a log point.

Lemma

Let \(W_Q = \Spec (Q\to k)\) be a log point and \(\theta:W_Q\xrightarrow{\sim} W_Q\) be an automorphism of the log structure \(M_W\) with \(\overline \theta = \id\) on \(\overline M_W\). Then these is a unique homomorphism \(\phi:Q^{gp}\to k^\times\) with \(\theta(q,c) = (q, \theta(q)c)\) for all \((q,c)\); conversely every such \(\phi\) defines such a \(\theta\).

Proof

The geometric data of \(\theta\) is unique, so take \(\theta:M_W\to M_W\) to be a morphism of the log structure only.

Now define \(\chi_\tau:T\to T_V\) by \(t\mapsto \phi_t\). The claim is then that \(\pi^T(y')\) lands in \(\mathcal Y^T_{\chi_\tau} = [V^{\chi_\tau}/T_V]\).

Backlinks 6