Deforming elephants of Q-Fano 3-folds Taro Sano Max Planck Institute for Mathematics

2. Results 1. Introduction and problems def

X : Q-Fano 3-fold = normal proj. 3-fold / C with only terminal singularities, −KX : ample Q-Cartier. Example 1.1. P3 , P(1, 1, 1, 2), some weighted c.i. 3-dim terminal sing: classified as follows. Fact 1.2. (Reid, Mori) p ∈ U : 3-dim. terminal sing. ⇒ • (U, p) : isolated hyperquot. sing, that is, U ' (f = 0)/Zr ( f ∈ OC4 : Zr -semi-invariant ). • ∃f : U → ∆1 : deformation of U ( ∆1 : small disk) s.t. Ut has only quotient sing (t 6= 0). f is called a Qsmoothing of U . Example 1.3. 1. U := C3 /Zr (1, a, r − a): quotient sing. (a, r: coprime). 2. U := (x2 + y 2 + z 2 + w2 = 0)/Z2 ⊂ C4 /Z2 (1, 1, 1, 0). Then U := (x2 + y 2 + z 2 + w2 + t)/Z2 → C is a Qsmoothing of U . (Ut has 2 × 21 (1, 1, 1)-sing.) Smooth Fano 3-folds: classified. Important fact in the classification; Fact 1.4. (Shokurov, Reid) X : Fano 3-fold with canonical Gorenstein sing. ⇒ ∃D ∈ |−KX | with only Du Val sing. Mukai’s classification of “indecomposable” Fano 3-fold w/ canonical Gor. sing. (Difficulties in non-Gorenstein case) • many possibilities of terminal sing. • 6 ∃D ∈ |−KX |: Du Val. Example 1.5. (1) X12,14 ⊂ P(2, 3, 4, 5, 6, 7) ⇒ |−KX | = ∅. (2) X15 = (z 5 +w13 +w23 +x15 +xy 7 = 0) ⊂ P(1, 2, 3, 5, 5) ⇒ |−KX | = {D}, D: non-Du Val. (3) X16 = ((yu2 + w4 ) + x(z 5 + zy 6 ) + x16 = 0) ⊂ P(1, 2, 3, 4, 7) ⇒ |−KX | = {D}, D: non-normal. Deformation theoretic approach to these difficulties; Conjecture 1.6. (Altınok-Brown-Reid) Let X : Q-Fano 3-fold. (1) X has a Q-smoothing, i.e. ∃X → ∆1 : deform. of X s.t. Xt has only quot sing. (2) Assume ∃D ∈ |−KX |. ⇒ ∃f : X → ∆1 : Q-smoothing of X s.t. ∃Dt ∈ |−KXt |: Du Val. (= simultaneous Q-smoothing).

2.1 Result for Conj (1) Theorem 2.1. (−) X: Q-Fano 3-fold. ⇒ ∃φ : X → ∆1 deformation of X s.t. Xt has only quot. sing. and A1,2 /4-sing. (A1,2 /4-sing.= (x2 + y 2 + z 3 + u2 = 0)/Z4 (1, 3, 2, 1). ) (Ingredients for the proof) Theorem 2.2. (−) X: Q-Fano 3-fold. ⇒ Deformations of X are unobstructed. Enough to find 1st order deform η ∈ TX1 . Analyze restriction map Π : TX1 → ⊕p∈Sing X TU1p . Problem: Π: NOT surjective in general. Consider local cohomology map τU ass’d to µ : U˜ → U : resolution and SNC exc. div. E. If U is Gorenstein, τU is τU : TU1 → HE2 (U˜ , Ω2U˜ (log E)(−E)). Proposition 2.3. (-) (U, p): germ of 3-dim term. sing. • τU = 0 ⇐⇒ (U, p): A1,2 /4-sing or quot. sing. 1 ∃ method to lift ηp ∈ T(U,p) s.t. τU 6= 0. (Due to Namikawa-Steenbrink, Minagawa)

Problem 2.4. ∃ non Q-smoothable Q-Fano 3-fold??

2.2 Result for Conj (2) Theorem 2.5. (−) X: Q-Fano 3-fold. Assume ∃D ∈ |−KX | with isolated sing. ⇒ X has a simult. Q-smoothing. (Ingredients for the proof) Fact 2.6. (Takagi) X: Q-Fano 3-fold s.t. ∃D0 ∈ |−KX |: normal at non-Gorenstein points of X. ⇒ ∃D ∈ |−KX |: normal. Enough to consider quot sing. and A1,2 /4-sing. I also need the following “good” weighted blow-up. Lemma 2.7. (U, p): germ of terminal quot. sing. or A1,2 /4-sing. D ∈ |−KU |: normal, but non Du Val. ⇒ ∃µ : U˜ → U : weighted blow-up s.t. ˜ − µ∗ (KU + D) < 0. KU˜ + D Remark 2.8. (Non-normal case): In klt case, Conj (2) is false. (e.g. X15 ⊂ P(1, 1, 5, 5, 7): general X15 has only non-normal elephants.) Problem 2.9. ∃ Q-Fano 3-fold s.t. h0 (X, −KX ) ≥ 2 and |−KX | contains only non-normal members?

BrAG 2014, University of Warwick