A classification of terminal quartic 3-folds and applications to rationality questions

Abstract

Abstract. This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Qfactorialisation of Y . In this case, the generators of Cl Y / Pic Y are "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces Y 4 ⊂ P 4 with rk Cl Y = 2 and show that when rk Cl Y ≥ 6, Y is always rational.

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

“…A rank one Fano 3-fold V 2g−2 of g ≥ 5 which contains a line and a conic also contains a rational normal cubic [38, 4.6 [45,46]. The classification is not yet complete, but already more than 200 families of rank two weak Fano 3-folds are known (with around 50 further cases still to be settled).…”

“…Even for the maximal defect σ = 15 it does not seem that the number of projective small resolutions of the Burkhardt quartic has been determined. (Recall it has at least one projective small resolution and exactly 2 45 3.5 × 10 13 Moishezon but not necessarily projective small resolutions).…”

“…Thanks to recent work of various authors -including Arap-Cutrone-Marshburn [2], Blanc-Lamy [8], Cutrone-Marshburn [18], Jahnke-Peternell-Radloff [40,41], Kaloghiros [45] and Takeuchi [93] -class (a) is known to consist of over 150 distinct deformation classes of semi-Fano 3-folds; many of these can be obtained by blowing up an appropriate smooth irreducible curve in an appropriate smooth rank one Fano 3-fold. This makes it relatively straightforward to determine many of the basic topological properties of such weak Fano 3-folds.…”

Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I

“…Note that the case g ≤ 4 is more difficult and a classification here must be very huge (cf. [Kal12]).…”

Rationality of an $S_6$-invariant quartic $3$-fold

“…(Although the proof also uses some computations carried in Section 2.) Unfortunately, we were not able to apply the results from [15], since non-rational X t all have defect equal 5 (see [1,Lemma 2]), which seems to contradict either [15,5.2,Lemma 8] or [15, 5.2, Proposition 3] (compare also with [15, Corollary 1] and the list of cases in [15,Main Theorem]).…”