Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On minimal varieties growing from quasismooth weighted hypersurfaces

Meng Chen, Chen Jiang, Binru Li

Abstract

This paper concerns the construction of minimal varieties with small canonical volumes. The first part devotes to establishing an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from which we construct plenty of new minimal $3$-folds including $59$ families of minimal $3$-folds of general type, several infinite series of minimal $3$-folds of Kodaira dimension $2$, $2$ families of minimal $3$-folds of general type on the Noether line, and $12$ families of minimal $3$-folds of general type near the Noether line. In the second part, we prove effective lower bounds of canonical volumes of minimal $n$-folds of general type with canonical dimension $n-1$ or $n-2$. Examples are provided to show that the theoretical lower bounds are optimal in dimension less than or equal to $5$ and nearly optimal in higher dimensions.

On minimal varieties growing from quasismooth weighted hypersurfaces

Abstract

This paper concerns the construction of minimal varieties with small canonical volumes. The first part devotes to establishing an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from which we construct plenty of new minimal -folds including families of minimal -folds of general type, several infinite series of minimal -folds of Kodaira dimension , families of minimal -folds of general type on the Noether line, and families of minimal -folds of general type near the Noether line. In the second part, we prove effective lower bounds of canonical volumes of minimal -folds of general type with canonical dimension or . Examples are provided to show that the theoretical lower bounds are optimal in dimension less than or equal to and nearly optimal in higher dimensions.

Paper Structure

This paper contains 26 sections, 18 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical volume and canonical dimension
  4. Kodaira dimension and numerical Kodaira dimension
  5. Cyclic quotient singularities
  6. The Reid basket
  7. Weighted projective spaces and weighted hypersurfaces
  8. Singularities on weighted hypersurfaces
  9. Weighted blow-ups of cyclic quotient singularities
  10. The nefness criterion
  11. Applications of the nefness criterion in constructing minimal varieties
  12. General construction
  13. Examples of minimal $3$-folds of general type with canonical volume less than $1$
  14. Examples of minimal $3$-folds of Kodaira dimension $2$
  15. Examples of minimal $3$-folds of general type near the Noether line
  16. ...and 11 more sections

Key Result

Theorem 1.3

Let $X=X^n_d\subset \mathbb{P}(b_1, \dots, b_{n+2})$ be an $n$-dimensional well-formed quasismooth general hypersurface of degree $d$ with $\alpha=d-\sum_{i=1}^{n+2}b_i>0$ where $b_1, \dots, b_{n+2}$ are not necessarily sorted by size. Denote by $x_1,\dots,x_{n+2}$ the homogenous coordinates of $\ma Then $K_Y$ is nef and $\nu(Y)\geq n-1$ where $\nu(Y)$ denotes the numerical Kodaira dimension.

Theorems & Definitions (55)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Rei87
  • Definition 2.5: Fle00
  • Definition 2.6: WPS, Fle00
  • Theorem 2.7: cf. Fle00
  • ...and 45 more