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Complete spelling rules for the Monster tower over three-space

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  • The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.

    Mathematics Subject Classification: 58A30, 58A17, 58K50.


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  • Figure 1.  The three critical planes $V, T_1$, and $T_2$, and their intersections, the distinguished lines $L_1, L_2$, and $L_3$.

    Figure 2.  Critical plane configurations that can appear in the distribution above an $R$ point (top left), a $V$ or $T_1$ point (top right), a $T_2$ point (bottom left), and an $L_j$ point (bottom right).

    Figure 3.  Critical plane configuration over $p_3\in RVL_1$. The left side shows the birth of $T_1(p_3)=\delta_2^1(p_3)$ as the first prolongation of the vertical plane at level 2. The right side shows the birth of $T_2(p_3)=\delta_1^2(p_3)$ as the second prolongation of the vertical plane at level 1. These two Baby Monsters meet in $\Delta^3$, and their intersection is the distinguished line $L_2(p_3)$. See Example 2.

    Figure 4.  Critical plane configuration over $p_4\in RVL_1T_2$. This shows the birth of $T_2(p_4)=\delta_1^3(p_4)$ as the third prolongation of the vertical plane at level 1. See Example 3.

    Table 1.  RVT Code Spelling Rules

    Letter Can be followed by Cannot be followed by
    $R$ $R, V$ $T_i, L_j$
    $V$ $R, V, T_1, L_1$ $T_2, L_2, L_3$
    $T_1$ $R, V, T_1, L_1$ $T_2, L_2, L_3$
    $T_2$ $R, V, T_2, L_3$ $T_1, L_1, L_2$
    $L_1$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$
    $L_2$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$
    $L_3$ $R, V, T_1, T_2, L_1, L_2, L_3$ $\emptyset$
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    Table 2.  Critical Hyperplane Configurations

    Last letter in RVT code of $p\in M^k$ Critical planes appearing in $\Delta^k(p)$
    $R$ $V$
    $V$ or $T_1$ $V$ and $T_1$
    $T_2$ $V$ and $T_2$
    $L_1, L_2,$ or $L_3$ $V, T_1$, and $T_2$
     | Show Table
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    Table 4.  Base Cases of Inductive Proof

    RVT code of $p_k\in M^k$ $T_{1}(p_k)$ $T_{2}(p_k)$
    $ \lambda V T_1^{m} L_1 T_{2} \, \, \text{for} \, \, m \geq 0$ None $\delta ^{m+3}_{k-m-3}(p_{k })$
    $ \lambda L_1 T_1^{m} L_1 T_{2} \, \, \text{for} \, \, m \geq 1$ None $\delta ^{m+3}_{k-m-3}(p_{k })$
    $ \lambda L_1 L_1 T_{2}$ None $\delta ^{3}_{k-3}(p_{k})$
    $ \lambda VT_1^{m} L_1L_{2} \, \, \text{for} \, \, m \geq 0$ $\delta ^{2}_{k-2}(p_{k})$ $\delta ^{m+3}_{k-m-3}(p_{k})$
    $ \lambda L_1T_1^{m} L_1L_{2} \, \, \text{for} \, \, m \geq 1$ $\delta ^{2}_{k-2}(p_{k})$ $\delta ^{m+3}_{k-m-3}(p_{k})$
    $ \lambda L_1L_1L_{2}$ $\delta ^{2}_{k-2} (p_{k})$ $\delta ^{3}_{k-3}(p_{k})$
    $ \lambda VT_1^{m} L_1 L_{3} \, \, \text{for} \, \, m \geq 0$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{m + 3}_{k-m-3}(p_{k})$
    $ \lambda L_1T_1^{m} L_1 L_{3} \, \, \text{for} \, \, m \geq 1$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{m + 3}_{k-m-3}(p_{k})$
    $ \lambda L_1L_1 L_{3}$ $\delta ^{1}_{k-1}(p_{k})$ $\delta ^{3}_{k-3}(p_{k})$
     | Show Table
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    Table 3.  Summary of Example 2: $RVL_1$

    Level $i$ Coordinates on $M^i$ $\mathbb P\Delta^{i-1} = F_i$ coordinates Critical planes in $\Delta^i$ RVT code of $p_i$
    $0$ $(x, y, z)$ n/anone n/a
    $1$ $(x, y, z, u_1, v_1)$
    $u_1=\frac{dy}{dx}, v_1=\frac{dz}{dx}$
    $[dx : dy : dz]$$V(p_1)=\delta_1^0$ $p_1=(p_0, l_0)\in R$
    $l_0 \subset \Delta^0=T_{p_0}M^0$
    $2$ $(x, y, z, u_1, v_1, u_2, v_2)$
    $u_2=\frac{dx}{du_1}, v_2=\frac{dv_1}{du_1}$
    $[dx : du_1 : dv_1]$$V(p_2)=\delta_2^0,$
    $p_2=(p_1, l_1)\in RV$
    $l_1 \subset V(p_1) \subset \Delta^1$
    $3$ $(x, y, z, u_1, v_1, u_2, v_3, u_3, v_3)$
    $u_3=\frac{du_1}{dv_2}, v_3=\frac{du_2}{dv_2}$
    $[du_1 : du_2 : dv_2]$$V(p_3)=\delta_3^0,$
    $p_3=(p_2, l_2)\in RVL_1$
    $l_2 = L_1(p_2) \subset \Delta^2$
     | Show Table
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