  In Chen-Cramer Crypto 2006 paper \cite{cc} algebraic geometric secret sharing
schemes were proposed such that the "Fundamental Theorem in
Information-Theoretically Secure Multiparty Computation" by Ben-Or, Goldwasser
and Wigderson \cite{BGW88} and Chaum, Cr\'{e}peau and Damg{\aa}rd \cite{CCD88}
can be established over constant-size base finite fields. These algebraic
geometric secret sharing schemes defined by a curve of genus $g$ over a
constant size finite field ${\bf F}_q$ is quasi-threshold in the following
sense, any subset of $u \leq T-1$ players (non qualified) has no information of
the secret and any subset of $u \geq T+2g$ players (qualified) can reconstruct
the secret. It is natural to ask that how far from the threshold these
quasi-threshold secret sharing schemes are? How many subsets of $u \in [T,
T+2g-1]$ players can recover the secret or have no information of the secret?
  In this paper it is proved that almost all subsets of $u \in [T,T+g-1]$
players have no information of the secret and almost all subsets of $u \in
[T+g,T+2g-1]$ players can reconstruct the secret when the size $q$ goes to the
infinity and the genus satisfies $\lim \frac{g}{\sqrt{q}}=0$. Then algebraic
geometric secret sharing schemes over large finite fields are asymptotically
threshold in this case. We also analyze the case when the size $q$ of the base
field is fixed and the genus goes to the infinity.
