[Paper Review] The dimension of jet schemes of singular varieties

Mircea Mustaţă|arXiv (Cornell University)|Apr 30, 2014

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper investigates the dimension of generalized jet schemes for singular algebraic varieties, focusing on how these dimensions reflect birational invariants like the log canonical threshold and minimal log discrepancy. It establishes a necessary condition—d^r ≤ n—for the r-fold iterated jet schemes of a homogeneous hypersurface to be pure-dimensional, linking singularity theory to algebraic geometry via motivic integration and jet scheme constructions.

ABSTRACT

Given a scheme X over a field k, a generalized jet scheme parametrizes maps from Spec(A) to X, where A is a finite-dimensional, local algebra over k. We give an overview of known results concerning the dimensions of these schemes when A has embedding dimension one, when they are related to invariants of singularities in birational geometry. We end with a discussion of more general jet schemes.

Motivation & Objective

• To understand how the dimensions of jet schemes encode information about singularities in algebraic geometry.
• To extend classical results on jet schemes to generalized jet schemes parametrized by local finite algebras.
• To explore the irreducibility and pure-dimensionality of iterated jet schemes for locally complete intersection varieties.
• To propose new invariants based on the asymptotic behavior of jet scheme dimensions across sequences of algebras of embedding dimension 2.
• To investigate whether the bound d^r ≤ n is sufficient for pure-dimensionality or irreducibility of r-fold jet schemes of homogeneous hypersurfaces with isolated singularities.

Proposed method

• Uses the universal property of jet schemes to define J_A(X) for any local finite k-algebra A, generalizing the standard jet schemes J_m(X) = J_{k[t]/(t^{m+1})}(X).
• Applies the change-of-variable formula in motivic integration to relate the dimensions of J_m(X) to invariants from log resolutions of pairs (Y,X).
• Employs projective limits to define the scheme of arcs J_{k[[t]]}(X) as the inverse limit of the J_m(X), crucial for motivic integration.
• Analyzes the fiber over the origin in jet schemes via the condition that f(u_1,…,u_n) = 0 for u_i in tR[t]/(t^{m+1}), linking vanishing to the degree d of the defining polynomial.
• Uses dimension estimates on subvarieties Z_j ⊂ J_{A_j}(X) defined by monomials of degree jd in r variables to derive bounds on n and d.
• Applies asymptotic analysis of binomial coefficients in the dimension formula to derive the inequality n ≥ d·∏_{i=1}^{r-1} (jd+i)/(j+i), leading to the limit condition d^r ≤ n.

Experimental results

Research questions

• RQ1 Under what conditions are the r-fold iterated jet schemes of a homogeneous hypersurface pure-dimensional?
• RQ2 Is the condition d^r ≤ n sufficient for the r-fold jet schemes of a homogeneous hypersurface with isolated singularity to be irreducible?
• RQ3 Can the irreducibility of iterated jet schemes be established for general homogeneous polynomials when d^2 ≤ n?
• RQ4 Is there a connection between the jet scheme dimension bounds and Lang’s C_r condition on fields?
• RQ5 Do the asymptotic dimensions of generalized jet schemes over algebras of embedding dimension 2 yield new invariants of singularities?

Key findings

• For a homogeneous hypersurface X ⊂ A^n defined by a degree d polynomial, the r-fold iterated jet schemes are pure-dimensional only if d^r ≤ n.
• If the r-fold jet schemes are irreducible, then the same inequality d^r ≤ n must hold, with strict inequality in the dimension estimate implying d^r < n.
• When d^2 ≤ n and the coefficients of the defining polynomial are algebraically independent over Q, the 2-fold iterated jet schemes are irreducible, as shown via F-singularities in positive characteristic.
• For uncountable fields, a very general homogeneous polynomial of degree d with d^2 ≤ n has irreducible 2-fold jet schemes.
• The bound d^r ≤ n arises from asymptotic analysis of binomial coefficients in the dimension formula for subvarieties of jet schemes.
• The condition d^r ≤ n is necessary for pure-dimensionality of r-fold jet schemes of homogeneous hypersurfaces with isolated singularities.