Dear Mukai-San,

maybe we can use your methods in some other contexts, e.g., Q-Fanos
or surfaces of general type.

Let's try a surface of gen type. Say p_g = 6, K^2 = 13, with K_S very
ample. Then S in PP^5 has degree 13, one more than the c.i. 2,2,3
and one less than the famous variety given by maximal Pfaffians of
a skew 7 x 7 matrix of linear forms. It's Hilbert series multiplied
up by (1-t)^6 is
  1 - t^2 - 4t^3 + 4t^4 + t^5 + t^7.
Thus one possible model for S is as the Pfaffians of a 5 x 5 matrix with
weights

  [ 1 1 1 2 ]
  [   1 1 2 ]
  [     1 2 ]
  [       2 ]

We can say that S is the intersection of PP^5 with the generic weighted
Grass(2,5) in PP(1^6,2^4), obtained by setting the 4 y_i equal to
quadratics in the 6 x_i. The generic Grassmann variety is a moduli space
for 2-forms of rank 2 in a 5-dimensional vector space with weights
1/2 times 1^4, 3 (say u_1,u_2,u_3,u_4,v, so that u_i \wedge u_j has weight 1
and u_i \wedge v has weight 2), up to weighted proportionality.

What structure on a variety gives a morphism to this wGrass? I think you
need a rank 2 vector bundle E with det E = Oh(1), 4 sections of E and one
section of E(1).