[Paper Review] On the existence of certain weak Fano threefolds of Picard number two

Maxim Arap, Joseph Cutrone+1 more|arXiv (Cornell University)|Dec 12, 2011

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper resolves the geometric realizability of 120 numerical cases of weak Fano threefolds of Picard number two with small anti-canonical maps, using blow-ups of curves on Fano threefolds of index 1, 2, or 3. It proves that 113 of the 120 open cases from [CM13] are geometrically realizable via Brill-Noether general K3 surfaces, while 12 remain unresolved due to technical obstructions.

ABSTRACT

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases.

Motivation & Objective

• To determine which numerical invariants of weak Fano threefolds of Picard number two with small anti-canonical maps are geometrically realizable.
• To close the gap in the classification of Sarkisov links between Fano threefolds of Picard number one by resolving open cases from [CM13].
• To establish geometric realizability of E1-E1 type Sarkisov links via blow-ups of curves on smooth Fano threefolds of index 1, 2, or 3.
• To identify and explain the 12 remaining unresolved cases due to technical obstructions in the current method.

Proposed method

• Construct weak Fano threefolds as blow-ups of smooth Fano threefolds of Picard number one along smooth irreducible curves of given degree and genus.
• Use the existence of Brill-Noether general K3 surfaces in Fano threefolds to ensure the freeness and nefness of the anti-canonical system.
• Apply Knutsen's classification of K3 surfaces in projective spaces to verify the non-existence of such surfaces in specific numerical cases.
• Use the criterion from [Knu13, Thm.3.2] to determine when a K3 surface with given (d,g) is Brill-Noether general.
• Verify that the anti-canonical system |−K_X| gives a small contraction by checking that the numerical cases do not appear in tables of divisorial contractions from [JPR05].
• Confirm that the flip morphism φ⁺ is of type E1 by ensuring the numerical cases do not appear in non-E1-E1 tables from [JPR11] or [CM13].

Experimental results

Research questions

• RQ1 Which numerical invariants of weak Fano threefolds of Picard number two with small anti-canonical maps are geometrically realizable?
• RQ2 Can the open cases in [CM13, Table E1-E1] be realized as blow-ups of curves on Fano threefolds of index 1, 2, or 3?
• RQ3 What conditions on degree d and genus g of a curve ensure that its blow-up yields a weak Fano threefold with a small anti-canonical map?
• RQ4 Why do 12 specific cases remain unresolved despite the general method?
• RQ5 Can the anti-canonical system |−K_X| be shown to give a small contraction rather than a divisorial one?

Key findings

• The numerical invariants in cases 44–46, 48, 70, 71, 86, 88, 97, 105 from [CM13, Table E1-E1] are geometrically realizable when starting from a smooth quadric threefold.
• The numerical invariants in cases 30, 34, 37, 41, 64, 84 are geometrically realizable when starting from a del Pezzo threefold of degree 4.
• The numerical invariants in cases 31, 35, 38, 40, 42, 65, 68, 69, 81, 83, 94, 96, 101 are geometrically realizable when starting from a del Pezzo threefold of degree 5.
• The numerical case 43 is not realizable due to the absence of a suitable K3 surface embedding.
• The numerical invariants in cases 3–8, 11–15, 18–20, 23, 55–57, 79, 93, 100, 104 are geometrically realizable when starting from a Fano threefold of index 1.
• The cases 16, 17, 21, 22, 24–26, 58, 60 are not realizable because no Brill-Noether general K3 surface exists with the required (d,g) in the corresponding Fano threefold.