[Paper Review] Two examples of surfaces with normal crossing singularities

Janós Kollár|ArXiv.org|May 7, 2007

Algebraic Geometry and Number Theory2 references4 citations

TL;DR

This paper constructs two irreducible, projective surfaces with normal crossing singularities that exhibit pathological behavior in their canonical line bundles: one whose canonical ring is not finitely generated, and another whose canonical bundle is not ample despite its pullback to the normalization being ample. These examples resolve a long-standing question in algebraic geometry regarding the behavior of pluricanonical systems on singular surfaces.

ABSTRACT

This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in [EGA,III.2.6.2] and raised again by Viehweg.

Motivation & Objective

• To construct explicit examples of normal crossing surfaces with unexpected behavior in their canonical line bundles.
• To resolve a question left open in EGA and posed by Viehweg regarding the ampleness of the canonical bundle on singular surfaces.
• To demonstrate that the canonical ring of a singular surface need not be finitely generated, even when the surface is projective and of general type.
• To illustrate that ampleness of the pullback to the normalization does not imply ampleness of the canonical bundle on the singular surface.
• To highlight the failure of canonical rings as birational invariants under flips in the minimal model program for semi-log-canonical surfaces.

Proposed method

• Constructing the first example, $T_1$, by gluing a surface $S_1$ along a curve via a finite morphism, resulting in a surface with two triple points.
• Using the normalization $ar{S}$ and the dualizing sheaf relation $n^* ho_S o ho_{ar{S}}(ar{D})$ to compute pluricanonical sections.
• Applying a gluing criterion: if a divisor $H$ on $S$ intersects the gluing curve transversally and is preserved under the gluing map, then its image is ample on the quotient.
• Computing sections of $ ho_S^{[m]}$ via the condition that sections on the normalization must be $(-1)^m au$-invariant along the singular locus.
• Using the fact that the canonical ring fails to be finitely generated when the gluing induces poles that cannot be canceled across branches.
• For the second example, $T_2$, constructing a quotient of a product of curves with involutions, leading to a surface with 24 nodes, and analyzing the behavior of the canonical bundle via local computations on the normalization.

Experimental results

Research questions

• RQ1 Can there exist a projective, irreducible surface with normal crossing singularities whose canonical ring is not finitely generated?
• RQ2 Is the canonical bundle of a singular surface with normal crossing singularities necessarily ample if its pullback to the normalization is ample?
• RQ3 How does the minimal model program affect the canonical ring of a singular fiber in a family of surfaces?
• RQ4 To what extent are pluricanonical systems preserved under flips when the singularities are not canonical?
• RQ5 What conditions on the gluing of curves ensure that sections of the dualizing sheaf descend to the singular surface?

Key findings

• There exists an irreducible, projective, normal crossing surface $T_1$ of general type whose canonical ring is not finitely generated.
• $T_1$ is constructed by gluing a surface $S_1$ along a curve via a finite morphism, with the canonical ring failing to be finitely generated due to incompatible pole orders across glued branches.
• The canonical bundle $ ho_{T_1}$ is not ample, but its pullback to the normalization is ample, showing that ampleness is not preserved under the singularization map.
• For the second example $T_2$, the canonical bundle $ ho_{T_2}$ is not ample, even though its pullback to the normalization is ample, resolving a question from EGA and Viehweg.
• Local computations on the normalization show that every section of $ ho_{T_2}^{2m}$ restricts to a section of $ ho_D^{2m}$ on the double curve $D$, forcing vanishing and thus non-ampleness.
• The surface $T_1$ is projective, as the gluing preserves ampleness under the transversality and invariance conditions on a suitable divisor $H$.