The Strong Monodromy Conjecture for a class of homogeneous polynomials in three variables

Abstract

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary number of variables $n$ and with $d$ denoting the degree of $f$, we characterize when $-n/d$ is a root of the Bernstein--Sato polynomial of $f$ in terms of elementary data involving logarithmic derivations. When we restrict to three variables, we prove the resulting class of polynomials satisfies the Strong Monodromy Conjecture, in the motivic sense.

Figures (6)

• Figure 1: The integral points of the triangular region $T_d$. The upper left corner is $(-d,d)$, the upper right $(0,d)$, the bottom right the origin. Many yet-to-be named integral points are drawn. The points $\phi_d(A_d), \phi_d(B_d), \phi_d(C_d),$ and $\phi_d(D_d)$ are labeled in the top left. The points $\{\phi_d(\alpha_{d,r})\}_r$ are the integral points on the line $u = 0$ and appear red. The points $\{\phi_d(\beta_{d,r})\}_r$ are the integral points on the line $u = -1$ and appear blue.
• Figure 2: A sketch of the projective zero locus of a polynomial in semi-simple standard provided$a_1 a_2 a_3 \neq 0$ and $2 \leq u < t$. The coordinate hyperplanes are red, the other components blue, and the intersections are $[0:0:1], [0:1:0], [1:0:0]$. When $u = 1$ (resp. $t-1$) the picture is similar except each blue component is now smooth at $[0:0:1]$ (resp. $[1:0:0]$). When $u = 0$, $f$ defines a decomposable hyperplane arrangement and the picture is different: the blue components are hyperplanes whose only common point is $[1:0:0].$
• Figure 3: The embedded resolution $\pi: F_0 \to F$ of $V_0 \subset F \simeq \mathop{\mathrm{\mathbb{P}}}\nolimits^2$ and the components of $\pi^{-1}(V_0)$; $L_x, L_y, L_z$ are the hyperplanes $\{x=0\}, \{y = 0\}, \{z=0\} \subset \mathop{\mathrm{\mathbb{P}}}\nolimits^2$, and the corresponding strict transforms are $E_x, E_y, E_z$.
• Figure 4: The rational map $\psi: \mathop{\mathrm{\mathbb{P}}}\nolimits^2 \dashrightarrow \mathop{\mathrm{\mathbb{P}}}\nolimits^1 = B$ appears in the bottom row; the points labeled $\times$ denote its indeterminancy locus. Here $\pi: F_0 \to \mathop{\mathrm{\mathbb{P}}}\nolimits^2$ resolves the indeterminancy locus, inducing $\tilde{\psi}: F_0 \to \mathop{\mathrm{\mathbb{P}}}\nolimits^1$. The blue $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$'s are some general fibers of $\tilde{\psi}$ which are sent to blue points in $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$; the chains of $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$'s are the special fibers of $\tilde{\psi}$, which are sent to the black points of $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$ (and represent $0$ and $\infty$). Note $E$ and $E^\prime$ biject onto $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$. Finally, $g: F_0 \to \Sigma$ contracts these chains of $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$'s to two single $\mathop{\mathrm{\mathbb{P}}}\nolimits^1$'s yielding the ruled surface $p: \Sigma \to \mathop{\mathrm{\mathbb{P}}}\nolimits^1$.
• Figure 5: The minimal embedded resolution of the two conics $D \subseteq \mathop{\mathrm{\mathbb{P}}}\nolimits^2 \simeq F$. The exceptional curves are black with labels $E_1, E_2, E_3, E_4$, whereas the two strict transforms are blue and unlabeled.

Theorems & Definitions (100)

• Theorem 1
• Proposition 1
• Theorem 2
• Definition 2
• Theorem 3
• Theorem 4
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4