[Paper Review] Multiplicity of jet schemes of monomial schemes

Cornelia Yuen|ArXiv.org|Jul 25, 2006

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper derives a closed-form formula for the multiplicity of jet schemes along irreducible components for reduced monomial hypersurfaces defined by $x_1\cdots x_r = 0$. Using induction and localization techniques, it shows that the multiplicity along the component corresponding to multi-index $(t_1,\ldots,t_r)$ with $\sum t_i = m+1$ is $\frac{(m+1)!}{t_1!\cdots t_r!}$, providing explicit scheme-theoretic information beyond irreducible decomposition.

ABSTRACT

This article studies jet schemes of monomial schemes. They are known to be equidimensional but usually are not reduced. We thus investigate their structure further, giving a formula for the multiplicity along every component of the jet schemes of a general reduced monomial hypersurface (that is, the case of a simple normal crossing divisor).

Motivation & Objective

• To determine the scheme-theoretic multiplicity of jet schemes along irreducible components for monomial hypersurfaces.
• To extend the understanding of jet schemes beyond their irreducible decomposition, particularly in the non-reduced case.
• To provide an explicit formula for multiplicities in the case of reduced monomial hypersurfaces (simple normal crossing divisors).
• To address a gap in the literature on explicit scheme structures of jet schemes, despite their importance in motivic integration and singularity theory.

Proposed method

• Reduces the problem to a localization at a minimal prime ideal $P(m;t_1,\ldots,t_r)$, using completion to preserve length.
• Applies a reduction step to eliminate variables with $t_i = 0$, reducing to the case where all $t_i > 0$.
• Uses the fact that the generators $g_0,\ldots,g_m$ form a regular sequence in the localized ring.
• Applies a key lemma: if $x_1\cdots x_r$ is a nonzerodivisor, then $\ell(R/(x_1\cdots x_r)) = \sum \ell(R/(x_i))$, enabling decomposition of the length computation.
• Establishes a ring isomorphism $S/(x_n^{(0)}) \cong R(m-1;t_1,\ldots,t_n-1,\ldots,t_r)/J_{m-1}(X)$ via variable shifting.
• Employs induction on $m$, using the base case $m=0$ where multiplicity is 1, and builds up to the general formula.

Experimental results

Research questions

• RQ1 What is the multiplicity of the $m$-th jet scheme of a reduced monomial hypersurface along each irreducible component?
• RQ2 How does the scheme structure of jet schemes of monomial schemes differ from their reduced structure?
• RQ3 Can a closed-form formula be derived for the multiplicity of jet schemes of monomial hypersurfaces?
• RQ4 How does the multiplicity depend on the multi-index $(t_1,\ldots,t_r)$ parameterizing the components?
• RQ5 What role does the multinomial coefficient $\frac{(m+1)!}{t_1!\cdots t_r!}$ play in the scheme-theoretic geometry of jet schemes?

Key findings

• The multiplicity of the $m$-th jet scheme $\mathcal{J}_m(X)$ along the irreducible component corresponding to $P(m;t_1,\ldots,t_r)$ is exactly $\frac{(m+1)!}{t_1!\cdots t_r!}$.
• The formula is proven by induction on $m$, with the base case $m=0$ yielding multiplicity 1.
• The multiplicity computation relies on the isomorphism $S/(x_n^{(0)}) \cong R(m-1;t_1,\ldots,t_n-1,\ldots,t_r)/J_{m-1}(X)$, which reduces the problem to the inductive hypothesis.
• The generators $g_0,\ldots,g_m$ form a regular sequence in the localized ring, ensuring the length computation is well-defined.
• The multiplicity formula matches the multinomial coefficient, reflecting combinatorial structure in the jet scheme's scheme-theoretic multiplicity.
• The result provides a complete scheme-theoretic description of jet schemes for reduced monomial hypersurfaces beyond their irreducible components.