Three Fano 3-folds with |-K| = \emptyset

Selma Alt{\i}nok and Miles Reid

In the context of classifying the Gorenstein rings over Q-Fano 3-folds,
especially those whose anticanonical ring has a small number of generators,
Alt{\i}nok's thesis (p. 31--32) contains the following plausible candidates
for Fano 3-folds with |-K| = \emptyset

Example 1. A^3 = -6 +11x1/2 +2/3 = 1/6
assuming generators PP(2,2,3,3,4,4,5,5) 
1 -3t^8-3t^9-3t^10
  +2t^12+6t^13+6t^14+2t^15
   -3t^17-3t^18-3t^19 +t^27

I believe that this is simply the codim 2 c.i. X_6,8 in PP(1,2,2,3,3,4)
divided out by Z/2 acting by
  (x,y_1,y_2,z_1,z_2,t) |-> (-x,-y_1,y_2,-z_1,z_2,-t).
This action interchanges the 6 x 1/2 points and the two 1/3 points of
X_6,8, and introduce 8 new 1/2 points from the fixed points of the action.
All the singularities, and invariants match up. The variety is essentially
just a complete intersection of two hypersurfaces in a cone over the
Veronese v_2(P^3). However, this is not codim 4; if you write out the
invariants of the group action, you find new generators in the same degree
as relations.

Example 2. A^3 = -6 +7x1/2 +2/3 +3.5/8 = (-60+16+45)/24 = 1/24
assuming generators PP(2,3,4,5,6,7,8,9)
1 -2t^12-t^13-2t^14-2t^15-t^16
  +2t^19+2t^20+3t^21+3t^22+2t^23+2t^24
   -t^27-2t^28-2t^29-t^30-2t^31 +t^43

If we write u,v,w for coordinates of P(1,3,5) and x,v,y,w,z,t for
coordinates of P(2,3,4,5,6,7), the general map P(1,3,5) to P(2,3.. 7) is
given by
  x = u^2, v = v, y = uv, w = w, z = uw, t = u^7.
i.e., we omit u, u^3, u^5 from k[u,v,w], so that this is an embedding. The
equations of the image E = P(1,3,5) are

       [ y  vx  z  wx   t  x^4 ]
 rank  [                       ]  <= 1
       [ v  y   w  z   x^3  t  ]

that is,

y^2 = v^2x, yw = vz, yz = vwx, yx^3 = vt, yt = vx^4, vxw = yz, vxz = wyx,
vx^4 = yt, vxt = yx^4, z^2 = w^2x, zx^3 = wt, zt = wx^4, wx^4 = zt,
wxt = zx^4, t^2 = x^7,

or deleting repeats,

y^2 = v^2x, yw = vz, yz = vwx, yx^3 = vt, yt = vx^4, z^2 = w^2x,
zx^3 = wt, zt = wx^4, t^2 = x^7,

These are equations of degree 8, 9, 10, 10, 11, 12, 12, 13, 14. So there
seems to be lots of scope for finding a general Y_12,14, reasonably
nonsingular, containing E. Next, the 2 forms of deg 10 through E
  yz = vwx, yx^3 = vt
define a pencil |10 A_Y - E|, so show that the canonical threshold can be
lowered. Maybe this is birationally a pencil of del Pezzo surfaces?

Example 3. A^3 = -6 +7x1/2 +4/5 +3.4/7 = (-175+56+120)/70 = 1/70
assuming generators PP(2,5,6,7,8,9,10,11)
1 -t^16-t^17-2t^18-t^19-2t^20-t^21-t^22
  +t^25+2t^26+2t^27+3t^28+3t^29+2t^30+2t^31+t^32
   -t^35-t^36-2t^37-t^38-2t^39-t^40-t^41 +t^57

A link starting with a Kamawata blowup would decrease A^3 by 1/2 or 1/20
or 1/70 (N.B. 3.4/7 means a singularity of type 1/7(1,2,5); we have to
write the inverses of 3,4 mod 7). Thus there is no link that is a simple
Type 1 projection. A simple-minded person might conclude that this variety
should be rigid.

I try to construct it by taking the finite projection to a hypersurface
\Vbar_48 in PP(2,5,6,7,8) (by omitting the last three coordinates
x_9,x_10,x_11. Following Ciliberto and Catanese's symmetric determinantal,
we could hope a resolution over R = k[PP(2,5,6,7,8)] of the form

0 --> R(-27)+R(-18)+R(-17)+R(-16) --M-->  R(-11)+R(-10)+R(-9)+R  --> 0

where M is a symmetric matrix with entries of degree

 [ 5  6  7  16]
 [            ]
 [ 6  7  8  17]
 [            ]
 [ 7  8  9  18]
 [            ]
 [16 17 18  27]

The determinant has degree 5 +7 +9 +17 = 48. Let B = column vector
[x_11, x_10, x_9, 1]. Then the equations of the ring are

 MB = 0 (the first three of degree 16, 17, 18)

and 6 others giving S^2(x_9,x_10,x_11) in terms of the minors of M
(of degrees 18, 19, 20, 20, 21, 22). Presumably the syzygies are also
accounted for.

Appendix
========
Cf. Selma's list of codimension 4 K3s ends with 24 "unsolved" cases that
have no Type 1 projection. Of these 22 have a Type 2 projection. The two
remaining cases have neither a Type 1 nor Type 2 projection, and are
roughly similar to the above elephantless Fano 3-folds.

Codimension 4 K3 surface, number 363, Altinok4(-22), with data
 D^2 = -4 + 1/2 + 2/3 + 2.3/5 + 2 x 5/6 = 1/30
 Weights: P(5,6,6,7,8,9,10)
 Numerator: t^51 -t^37 -t^36 -2t^35 -t^34 -2t^33 -t^32 -t^31 +t^29
+2t^28 +2t^27 +3t^26 +3t^25 +2t^24 +2t^23 +t^22 -t^20 -t^19
-2t^18 -t^17 -2t^16 -t^15 -t^14 +1
 Basket: [ 2, 1 ], [ 3, 1 ], [ 5, 2 ], [ 6, 1 ], [ 6, 1 ]

Codimension 4 K3 surface, number 367, Altinok4(-19), with data
 D^2 = -4 + 2 x 1/2 + 2 x 2/3 + 5/6 + 6/7 = 1/42
 Weights: P(6,6,7,8,9,10,11)
 Numerator: t^57 -t^41 -t^40 -2t^39 -t^38 -2t^37 -t^36 -t^35 +t^32
+2t^31 +2t^30 +3t^29 +3t^28 +2t^27 +2t^26 +t^25 -t^22 -t^21
-2t^20 -t^19 -2t^18 -t^17 -t^16 +1
 Basket: [ 2, 1 ], [ 2, 1 ], [ 3, 1 ], [ 3, 1 ], [ 6, 1 ], [ 7, 1 ]

There are two codimension 3 examples with no Type 1 or Type 2 projections;

Codimension 3 K3 surface, number 59, Altinok3(61), with data
 Weights: P(5,5,6,7,8,9)
 Numerator: -t^40 +t^26 +t^25 +t^24 +t^23 +t^22 -t^18 -t^17 -t^16
-t^15 -t^14 +1
 Basket: [ 5, 1 ], [ 5, 2 ], [ 5, 2 ], [ 6, 1 ]

Codimension 3 K3 surface, number 84, Altinok3(70), with data
 Weights: P(5,6,7,8,9,10)
 Numerator: -t^45 +t^29 +t^28 +t^27 +t^26 +t^25 -t^20 -t^19 -t^18
-t^17 -t^16 +1
 Basket: [ 2, 1 ], [ 3, 1 ], [ 5, 1 ], [ 5, 2 ], [ 7, 1 ]

Codimension 3 K3 surface, number 86, Altinok3(65), with data
 Weights: P(4,5,5,6,7,8)
 Numerator: -t^35 +t^23 +t^22 +t^21 +t^20 +t^19 -t^16 -t^15 -t^14
-t^13 -t^12 +1
 Basket: [ 2, 1 ], [ 4, 1 ], [ 5, 1 ], [ 5, 1 ], [ 5, 2 ]