[Paper Review] On minimal varieties growing from quasismooth weighted hypersurfaces

Meng Chen, Chen Jiang+1 more|arXiv (Cornell University)|May 20, 2020

Algebraic Geometry and Number Theory35 references4 citations

TL;DR

This paper establishes a new nefness criterion for the canonical divisor of weighted blow-ups over quasismooth weighted hypersurfaces, enabling the construction of 59 families of minimal 3-folds of general type, several infinite series of Kodaira dimension 2 3-folds, and examples near the Noether line. The method yields effective lower bounds on canonical volumes for $ n $-folds of general type with canonical dimension $ n-1 $ or $ n-2 $, shown to be optimal in dimensions ≤5.

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.

Motivation & Objective

• To develop an effective nefness criterion for the canonical divisor of a weighted blow-up over a quasismooth weighted hypersurface with a single non-canonical singularity.
• To construct new families of minimal 3-folds of general type and Kodaira dimension 2 using this criterion.
• To establish effective lower bounds on the canonical volume of minimal $ n $-folds of general type with canonical dimension $ n-1 $ or $ n-2 $.
• To show that the theoretical lower bounds are optimal in dimensions at most 5 and nearly optimal in higher dimensions.
• To provide explicit examples that realize the theoretical bounds, particularly near the Noether line in dimension 3.

Proposed method

• Apply weighted blow-up at a non-canonical cyclic quotient singularity $ Q $ of a well-formed quasismooth weighted hypersurface $ X o ext{Proj}(b{C}[x_1, ext{...},x_{n+2}]) $ with weights $ (b_1, ext{...},b_{n+2}) $.
• Derive a sufficient condition for the canonical divisor $ K_Y $ on the blow-up $ Y $ to be nef using inequalities involving the degree $ d $, weights $ b_j $, and singularity invariants $ r, e_j $.
• Use the condition $ ext{Vol}(Y) o ext{Vol}(X) imes ext{adjustment factor from blow-up} $ to estimate canonical volume via intersection theory.
• Employ the numerical criterion $ (K_Y imes C) < 0 $ to derive contradiction under assumptions, proving $ K_Y $ is nef.
• Use the structure of the exceptional divisors $ E_1, E_2 $ and their intersections with strict transforms of hyperplane sections to verify positivity of $ K_Y^{n-1} imes E_1 $.
• Verify irreducibility of general hypersurfaces in $ b{P}(b_n, b_{n+1}, b_{n+2}) $ and well-formedness of the weighted projective space $ b{P}(e_1, ext{...},e_n) $ to ensure geometric consistency.

Experimental results

Research questions

• RQ1 Can a systematic construction of minimal 3-folds of general type be achieved via weighted blow-ups on quasismooth weighted hypersurfaces?
• RQ2 What conditions ensure that the canonical divisor of the blow-up remains nef?
• RQ3 What are effective lower bounds for the canonical volume of minimal $ n $-folds of general type with canonical dimension $ n-1 $ or $ n-2 $?
• RQ4 Are these theoretical lower bounds optimal or nearly optimal in low and higher dimensions?
• RQ5 Can new families of minimal 3-folds be constructed that differ from known examples in deformation invariants and Picard number?

Key findings

• The paper constructs 59 new families of minimal 3-folds of general type, all with Picard number at least 2 and distinct from Iano-Fletcher’s lists.
• It provides several infinite series of minimal 3-folds of Kodaira dimension 2, arising from weighted blow-ups at singular points of quasismooth hypersurfaces.
• Two families of minimal 3-folds of general type lie exactly on the Noether line, achieving the minimal possible canonical volume for that bound.
• Twelve additional families of minimal 3-folds of general type are constructed near the Noether line, approaching the theoretical minimum volume.
• Effective lower bounds for canonical volumes of minimal $ n $-folds of general type with canonical dimension $ n-1 $ or $ n-2 $ are proven, and shown to be optimal in dimensions ≤5.
• Examples are constructed that realize the theoretical lower bounds, confirming their sharpness in dimensions at most 5 and near-optimality in higher dimensions.