Monodromy and vanishing cycles for complete intersection curves

Abstract

We compute the topological monodromy of every family of complete intersection curves. Like in the case of plane curves previously treated by the second-named author, we find the answer is given by the $r$-spin mapping class group associated to the maximal root of the adjoint line bundle. Our main innovation is a suite of tools for studying the monodromy of sections of a tensor product of very ample line bundles in terms of the monodromy of sections of the factors, allowing for an induction on (multi-)degree.

Figures (7)

• Figure 1: Exhibiting $\beta$ with $i(\beta,\alpha_{i+1}) = 0$ in $\Gamma_{i+1}$. The arc $\alpha_{i+1}$ is shown in its entirety in red; $\beta$ is depicted schematically in blue. Top row: if necessary, apply a half-twist supported in $S_i$ so that $\alpha_{i+1}$ and $\beta$ share exactly one endpoint. Bottom row: it may be necessary to switch the common endpoint, so that both arcs cross the handle in the same direction when starting at their unique common endpoint. Then a further sequence of half-twists in $\Gamma_i$, followed by a half-twist about $\alpha_{i+1}$, take $\beta$ to an arc supported on $S_i$.
• Figure 2: Inductively exhibiting $\beta$ with $k(\beta) = 1$ in $\Gamma_{i+1}$. As before, $\alpha_{i+1}$ is shown in red and $\beta$ is shown schematically in blue. $\gamma$ is shown schematically in green. The top and bottom rows depict the two possibilities: either $\gamma$ goes through the new handle, or it does not.
• Figure 3: The inductive step: decreasing $k(\beta)$. As before, $\beta$ is shown in blue and $\gamma$ is shown in green.
• Figure 4: Illustrating the construction in the case of $D$ an elliptic curve represented as a $3$-sheeted simple branched covering. First panel: a schematic picture of the branched covering, with branch cuts indicated as dashed line segments. Second panel: selecting the first $N-1 = 2$ arcs so as to span a disk. Third panel: each additional arc attaches a $1$-handle to the existing surface. Fourth panel: When all arcs have been added, the neighborhood fills $D$ away from a union of disks. A simple rendering of the neighborhood is included in the lower-right corner of panels 2-4.
• Figure 5: Three points of view on the tacnode singularity. Top: A vanishing cycle associated to the tacnodal singularity $w(w-z^2)$. In the plot, $s_0 = 0.05$, making the critical fiber occur for $t_0 = 2 \sqrt{s_0} \approx 0.447$. The figure shows the real points of $E^{loc}_{s_0,t}$ for $t = 1, 0.5, 0.451$ from left to right. $D$ appears as the line (orange), and $C_t$ is the parabola (green) of varying heights. Middle: a topological picture of $E^{loc}_{s_0,t_0}$. The left (green) half is $\widetilde{C}_{t_0}$, the right is $\widetilde{D}$, and these are joined along the curves $\beta_i, \beta_j$. $a$ appears as the curve shown in the middle. Bottom: representing $\widetilde{D}$ as a branched covering of $\mathbb{CP}^1$ via the pencil map for $C_t$. Observe that the path $\tau \subset \mathbb{CP}^1$ has distinguished lift $a \cap \widetilde{D}$.

Theorems & Definitions (118)

• Theorem 1
• Remark 1.1
• Theorem 2
• Remark 1.2
• Remark 1.3
• Corollary 3
• Remark 1.4
• Remark 1.5: A note on terminology and notation
• Conjecture 1.6
• Lemma 2.1: Multidegree-genus formula