Proposition 3. .3.2. Let µ be a p-restricted dominant weight and µ' a dominant weight for SL(n). If V (µ)|G is a composition factor of V (µ')|G then ƒ (µ) ≤ ƒ (µ'). Σ i=1 Σ

Proposition 3. For any domain V , aV ba can be UC-realized with statistical security in the a-smt- hybrid model, in expected-constant rounds and against an adaptive and malicious t-adversary, pro- vided t < n .

Proposition 3. If there exists a κ∗ ∈ [0, 1] such that ΠT1(κ∗) + ΠT1(κ∗) + (ΠT0(κ∗) − ΠT0(κ∗)) = ΠD0(κ∗), then under a Xxxx-Bargaining solution for the licensing fee, the threat to launch an AG is credible for all κ ≥ κ∗ if ∂ΠT1/∂κ ≈ 0. Proof. See Appendix B. The condition ∂ΠT1/∂κ ≈ 0 – that the equilibrium profit for the branded firm in T1 is nearly constant, i.e., not changing with the relative first mover advantage between generic one and generic two – is stronger than needed. What we need for net surplus to be increasing in κ is the condition |∂ΠT1/∂κ| < |∂ΠT1/∂κ|, i.e., the branded firm’s equilibrium profit is decreasing in first mover advantage at a slower rate than the increase in the equilibrium profit of the first generic entrant so that the overall net surplus still keeps on increasing in κ (recall that ΠT0 does not change with κ, ˜

Proposition 3. .1.2. If X is a cubic surface and L1, . . . , L6 is any subset of six mutually skew lines chosen among the 27 lines as in corollary 3.1.1. Then there is a morphism πj : X → P2, making X isomorphic to the blow-up of P2 along six points x1j , . . . , x6j that L1, . . . , L6 are the exceptional curves for πj. such As in Theorem 2.4.4, we know that Br X/ Br k →−∼ H1(Gk, Pic X) is one of {0}, Z/2Z, (Z/2Z)2, Z/3Z or (Z/3Z)2. A more explicit description of the elements of H1(Gk, Pic X) for X a cubic surface is given by Xxxxxxxxxx-Xxxx [Swi93] and describes specific subsets or partitions of Pic X which correspond to the possible non-trivial elements.

Proposition 3. [FM00] Consider a set of 3 players. In the model with a com- plete, synchronous network of pairwise authenticated channels weak 2-cast (i.e., weak broadcast) unconditionally secure against one corrupted player implies ef- ﬁcient 2-cast unconditionally secure against any number of corrupted players. Hence, as follows from the previous propositions, in order to show that global broadcast or MPC are eﬃciently achievable for t < n/2, it is suﬃcient to show that the simulation of weak 2-cast among any set of three players is possible that is unconditionally secure against one corrupted player. This fact is captured by the following proposition.