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Singularity of $\mathbb{Q}$-divisors of multidegree one in multiprojective space

Supravat Sarkar

TL;DR

We address singularities of effective $\mathbb{Q}$-divisors on multiprojective spaces $X=\prod_{i=1}^n \mathbb{P}^{m_i}$ of multidegree $(1,...,1)$. The main result shows that if $\Delta\ge 0$ has multidegree $(1,...,1)$ and $\lfloor\Delta\rfloor=0$, then $(X,\Delta)$ is klt, and if $\lfloor\Delta\rfloor$ is irreducible, then $(X,\Delta)$ is plt. For irreducible divisors, inversion of adjunction yields that such divisors have rational singularities, and the authors derive a lower bound on the log canonical threshold for irreducible hypersurfaces of multidegree $(d,...,d)$. The work extends known results on square-free polynomials to multiprojective spaces, connects to the minimal model program via connectedness and adjunction, and generalizes Bath–Mustata–Walther-style phenomena as a corollary.

Abstract

We study singularity of effective $\mathbb{Q}$-divisors on products of projective spaces of multidegree $(1,1...,1).$ This generalizes works of Bath, Musta{ţ}{ă} and Walther on singularity of square-free polynomials. We also give a lower bound on the log canonical threshold of a hypersurface in products of projective spaces.

Singularity of $\mathbb{Q}$-divisors of multidegree one in multiprojective space

TL;DR

We address singularities of effective -divisors on multiprojective spaces of multidegree . The main result shows that if has multidegree and , then is klt, and if is irreducible, then is plt. For irreducible divisors, inversion of adjunction yields that such divisors have rational singularities, and the authors derive a lower bound on the log canonical threshold for irreducible hypersurfaces of multidegree . The work extends known results on square-free polynomials to multiprojective spaces, connects to the minimal model program via connectedness and adjunction, and generalizes Bath–Mustata–Walther-style phenomena as a corollary.

Abstract

We study singularity of effective -divisors on products of projective spaces of multidegree This generalizes works of Bath, Musta{ţ}{ă} and Walther on singularity of square-free polynomials. We also give a lower bound on the log canonical threshold of a hypersurface in products of projective spaces.
Paper Structure (3 sections, 6 theorems, 3 equations)

This paper contains 3 sections, 6 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Acknowledgement

Key Result

Theorem A

Let $n\geq1$, $m_1, m_2,..., m_n$ be positive integers, $X=\prod_{i=1}^n\mathbb{P}^{m_i}$. Let $\Delta\geq0$ be a $\mathbb{Q}$-divisor in $X$ of multidegree $(1,1,...,1).$ Then

Theorems & Definitions (13)

  • Theorem A
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • ...and 3 more