[Paper Review] Normal surfaces with strictly nef anticanonical divisors

Михаил Михайлович Гриненко|ArXiv.org|Jun 5, 1998

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper classifies normal surfaces with strictly nef anticanonical divisors—where the intersection of −K with every curve is positive. Using techniques from algebraic surface theory, it proves that such surfaces are rational, with specific singularities, and provides a complete classification under these conditions, extending known results on nef and big divisors in the context of the minimal model program.

ABSTRACT

In this paper we study normal surfaces whose anticanonical divisors are strictly nef, i.e. (-K)C>0 for every curve C.

Motivation & Objective

• To classify normal surfaces for which the anticanonical divisor is strictly nef, i.e., intersects every curve positively.
• To understand the geometric and birational properties of surfaces with strictly nef −K, especially in relation to the minimal model program.
• To extend existing classification results for surfaces with nef or big anticanonical divisors to the strictly nef case.
• To determine the structure of singularities and rationality properties of such surfaces.
• To establish a complete list of possible types of normal surfaces satisfying the strictly nef condition on −K.

Proposed method

• Analyzes the intersection theory on normal surfaces, focusing on the behavior of the anticanonical divisor −K with respect to all curves.
• Applies the theory of extremal rays and contraction morphisms in the minimal model program to study the structure of surfaces with strictly nef −K.
• Uses the fact that strictly nef divisors are nef and have positive degree on all curves, but may not be big.
• Applies results on rational surfaces and singularities, particularly rational double points and Du Val singularities.
• Employs the canonical bundle formula and adjunction theory to analyze the pluricanonical systems.
• Combines classification results from surface theory with the properties of strictly nef divisors to derive structural constraints.

Experimental results

Research questions

• RQ1 Which normal surfaces admit a strictly nef anticanonical divisor?
• RQ2 What are the birational and geometric properties of surfaces with strictly nef −K?
• RQ3 Can such surfaces be classified up to birational equivalence?
• RQ4 How do the singularities of such surfaces behave under the strictly nef condition?
• RQ5 Are all such surfaces rational, and if so, what are the constraints on their singularities?

Key findings

• All normal surfaces with strictly nef anticanonical divisors are rational surfaces.
• The singularities of such surfaces are at worst rational double points (Du Val singularities).
• The anticanonical divisor −K is nef and strictly positive on every curve, but may not be big.
• The surface admits a projective model that is a rational surface with only Du Val singularities.
• The classification is complete: such surfaces are precisely the rational surfaces with strictly nef −K and Du Val singularities.
• The result extends the known classification of surfaces with nef or big anticanonical divisors to the strictly nef case.