Ample vector bundles and introduction to Mori theory Fall 2014

Problem set V, for November 13th All varieties are defined over an algebraically closed field k. Recall the following basic notions which can be found in the Hartshorne’s book. Differentials. Let A be a finitely generated k algebra (assume it is a domain). The A module ΩA is generated by symbols da, with a ∈ A modulo the relations: (1) d(a + b) = da + db, (2) d(ab) = adb + bda, (3) d|k = 0. On a variety X we define the coherent sheaf ΩX by glueing its values on the affine pieces. L d Coherent sheaves on the projective space. Let S = d≥0 S = k[x0, . . . , xn] be the homogeneous coordinate ring of Pn = Proj S which is covered by n affine sets Ui ' Ak with coordinates k[xo/xi, . . . , xn/xi]. Recall that for any coherent sheaf F on Pn we associate a graded S-module of its sections Γ∗(F) = L H0( n, F(j)). On the other hand, for a graded S-module j∈Z P M = L M j we define a quasi-coherent sheaf M on n such that on every j∈Z f P affine set U we have M(U ) = M 0 = {m/xr : m ∈ M r}. i f i xi i

1. Euler sequence [Hartshorne, II.8]. In the notation introduced above, consider a morphism of graded S modules

n M S(−1) · vj −→ S j=0

where vj’s are generators of degree 1 and each vj is mapped to xj. Let M be the kernel of this map.

(a) Show that we get the exact sequence of O sheaves

0 −→ Mf −→ O(−1)⊕n+1 −→ O −→ 0

(b) On Ui with coordinate ring k[x0/xi, . . . , xn/xi] we define a map ψ :Ω → O(−1)⊕n+1 by setting ψ (d(x /x )) = (x v − x v )/x2. i Ui |Ui i r i i r r i i

Prove that the map is isomorphism onto Mf|Ui . (c) Prove that the homomorphisms ψi glue to isomorphism of sheaves

ΩPn → Mf so that we have the Euler sequence

⊕n+1 0 −→ ΩPn −→ O(−1) −→ O −→ 0

2. Find the connecting (boundary) homomorphism H0(Pn, O) → 1 n H (P , ΩPn ) in the sequence of cohomology of the Euler sequence; cal- ˇ culate it using the covering (Ui) and express it in terms of Cech coho- mology.

3. For d > 0 consider the twisted Euler sequence

⊕n+1 0 −→ ΩPn ⊗ O(d) −→ O(d − 1) −→ O(d) −→ 0 Prove that the H0 of this sequence is exact and that it actually splits P ⊕(n+1) by O(d) 3 s → (∂s/∂xi) · vi ∈ O(d − 1) .

p 4. Use Euler sequence to calculate cohomology of sheaves Ω n of p-forms P on Pn. Hint: prove the following lemma. Given an exact sequence of vector bundles 0 → V → W → L → 0, with L of rank 1, for any positive p we get the exact sequence 0 → ΛpV → ΛpW → Λp−1V ⊗L → 0.

5. Prove the relative version of the Euler sequence: Let E be a vector bundle over smooth variety X. Consider its projectivisation p : P(E) → X. By ΩP(E)/X we denote the sheaf of relative differentials which are in ∗ the kernel of the map Dp : p ΩX → ΩP(E). Then we have the following relative Euler sequence on P(E):

∗ 0 −→ ΩP(E)/X −→ p E ⊗ OP(E)(−1) −→ O −→ 0 Use the relative Euler sequence to calculate the canonical divisor of

P(E) in terms of KX , detE, rkE and OP(E)(1).