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The minimal exponent of cones over smooth complete intersection projective varieties

Qianyu Chen, Bradley Dirks, Mircea Mustaţă

TL;DR

The paper extends the notion of the minimal exponent $\widetilde{\alpha}(Z)$ from hypersurfaces to local complete intersections defined by a regular sequence of homogeneous polynomials, under a transversality assumption. It provides an upper bound in the weighted homogeneous setting and a general lower bound via strong factorizing resolutions à la Bravo–Villamayor, then proves an explicit formula for the minimal exponent in the homogeneous complete-intersection case with possibly different degrees. The main result shows $\widetilde{\alpha}(Z)=\min\{ i+\frac{1}{d_i}(n-\sum_{j=1}^i d_j) : 1\le i\le r\}$ (with a prescribed index $p$) and establishes the equality by constructing an explicit strong factorizing resolution and matching bounds. This yields precise criteria linking the minimal exponent to the log canonical threshold and rational singularities in the homogeneous setting, and broadens the toolkit for singularity theory and birational geometry via Bernstein–Sato and Hodge-theoretic methods.

Abstract

We compute the minimal exponent of the affine cone over a complete intersection of smooth projective hypersurfaces intersecting transversely. The upper bound for the minimal exponent is proved, more generally, in the weighted homogeneous setting, while the lower bound is deduced from a general lower bound in terms of a strong factorizing resolution in the sense of Bravo and Villamayor.

The minimal exponent of cones over smooth complete intersection projective varieties

TL;DR

The paper extends the notion of the minimal exponent from hypersurfaces to local complete intersections defined by a regular sequence of homogeneous polynomials, under a transversality assumption. It provides an upper bound in the weighted homogeneous setting and a general lower bound via strong factorizing resolutions à la Bravo–Villamayor, then proves an explicit formula for the minimal exponent in the homogeneous complete-intersection case with possibly different degrees. The main result shows (with a prescribed index ) and establishes the equality by constructing an explicit strong factorizing resolution and matching bounds. This yields precise criteria linking the minimal exponent to the log canonical threshold and rational singularities in the homogeneous setting, and broadens the toolkit for singularity theory and birational geometry via Bernstein–Sato and Hodge-theoretic methods.

Abstract

We compute the minimal exponent of the affine cone over a complete intersection of smooth projective hypersurfaces intersecting transversely. The upper bound for the minimal exponent is proved, more generally, in the weighted homogeneous setting, while the lower bound is deduced from a general lower bound in terms of a strong factorizing resolution in the sense of Bravo and Villamayor.
Paper Structure (4 sections, 6 theorems, 50 equations)

This paper contains 4 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. An upper-bound in the weighted homogeneous case
  3. A general lower bound via a strong factorizing resolution
  4. The formula in the homogeneous case

Key Result

Theorem 1.1

Let $f_1,\ldots,f_r\in {\mathbf C}[x_1,\ldots,x_n]$ be homogeneous polyomials that form a regular sequence, with ${\rm deg}(f_i)=d_i$ for $1\leq i\leq r$, and such that $2\leq d_1\leq\ldots\leq d_r$. For every $i$, we denote by $H_i$ the hypersurface defined by $f_i$ in ${\mathbf A}^n$ and by $Z$ th where $p$ is the smallest $i\leq r$ that satisfies $d_1+\ldots+d_i>n$ (with the convention that $p=

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm_upper_bound']}
  • proof : Proof of Theorem \ref{['thm_lower_bound']}
  • Proposition 4.1
  • ...and 4 more