{"id":175,"date":"2010-07-12T13:26:39","date_gmt":"2010-07-12T13:26:39","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=175"},"modified":"2012-02-27T12:57:42","modified_gmt":"2012-02-27T12:57:42","slug":"rank-2-fanos","status":"publish","type":"page","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=175","title":{"rendered":"Rank 2 Fano 3-folds"},"content":{"rendered":"

We compute the quantum period sequences of Fano 3-folds in the Mori-Mukai list.<\/p>\n

    \n
  1. [Not very Fano]The blow-up of V_1 with centre an elliptic curve which is the intersection of two members of\u00a0 |{-{1\/2}} K|. This is a hypersurface in a toric variety F. \u00a0 The divisor diagram for F is \\begin{array}{l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 2 & 3 \\end{array}.
    \nNote that F is a scroll over \\CC P^1 with fibre \\CC P(1,1,2,3). There is a morphism F \\to \\CC P(1,1,1,2,3) , which is the blow up along x_0=x_1=0; this map sends [s_0,s_1,x, x_2, y, z] to [s_0x,s_1x,x_2,y,z]. There are two line bundles L, M on F: s_0,s_1 are section of L; xs_0, xs_1,x_2 are sections of M; y is a section of 2M and z of 3M. X is cut out by a section of 6M: we have -K_X=L+M.<\/p>\n

    Applying quantum Lefschetz gives:
    \nI_X= \\sum_{l, m\\geq 0} t^{l+m}\\frac{(6m)!}{l!l!m!(2m)!(3m)!\\Gamma(1+m-l)}.
    \nRegularizing this (that is, pre-multiplying by e^{-61t} so as to kill the linear term in t, and then replacing \\sum a_kt^k by \\sum k! a_k t^k) gives a period sequence that is not in our list:
    \n1+10380 t^2+2082840 t^3+650599740 t^4+199351017360 t^5\\cdots<\/p>\n

    Note that there are two birational models of the ambient space here, corresponding to the two chambers in the divisor diagram.\u00a0 But the sum defining the I-function really takes place over the intersection of the Mori cones of the two birational models, because of the factor of 1\/\\Gamma(1+m-l) in the summand.\u00a0 This factor vanishes outside one of the Kahler cones; similarly 1\/l! = 1\/\\Gamma(1+l) vanishes outside the other Kahler cone.\u00a0 Similar things happen in many of the examples below.<\/span><\/span><\/li>\n

  2. [Not very Fano]The double cover of \\CC P^1 \\times \\CC P^2 branched along a divisor of bidegree (2,4). This is\u00a0 a hypersurface in a toric variety F with divisor diagram \\begin{array}{l l l l l l} 1 & 1 & 0 & 0 & 0 & 1\\\\ 0 & 0 & 1 & 1 & 1 & 2 \\end{array}. With coordinates x_0, x_1,y_0,y_1,y_2 ,z, the defining equation of X is z^2=f_{2,4}(x_0,x_1;y_0,y_1,y_2). Denote by L the line bundle with sections x_i and by M the line bundle with sections y_j; then -K_X = L+M.Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(2l+4m)!}{l! l! m! m! m! (l+2m)!}.
    \nRegularizing this gives a period sequence that is not in our list:
    \n1+470 t^2+21216 t^3+1562778 t^4+114717120 t^5+\\cdots<\/li>\n
  3. [Not very Fano]the blow up of V_2 with centre an elliptic curve which is the intersection of two elements of |-1\/2K|.\u00a0 This is a hypersurface in a toric variety F. The divisor diagram for F is \\begin{array}{l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 1 & 2 \\end{array}.
    \nNote that F is a scroll over \\CC P^1 with fibre \\CC P(1,1,1,2). There is a morphism F \\to \\CC P(1,1,1,1,2) , which is the blow up along x_0=x_1=0; this map sends [s_0,s_1,x, x_2, x_3, y] to [s_0x,s_1x,x_2,x_3,y]. There are two line bundles L, M on F: s_0,s_1 are section of L; xs_0, xs_1,x_2, x_3 are sections of M; y is a section of 2M. X is cut out by a section of 4M: we have -K_X=L+M. Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(4m)!}{l! l! m! m! (2m)! \\Gamma (1+m-l)}.
    \nRegularizing this gives a period sequence that is not in our list:
    \n1+300 t^2+8472 t^3+438588 t^4+21183120 t^5+\\cdots<\/li>\n
  4. the blow up of P^3 with centre an intersection of two cubics. Thus, X is a divisor of bidegree (1,3) on \\CC P^1 \\times \\CC P^3. We denote by L and M the pull backs of the tautological bundles on the two factors.\u00a0 We have -K_X = L + M, and quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+3m)!}{l! l! m! m! m! m!}.
    \nRegularizing this gives period sequence 49:
    \n1+90 t^2+1518 t^3+46086 t^4+1327320 t^5+\\cdots<\/li>\n
  5. the blow up of V_3 with centre a plane cubic.This is a hypersurface in a toric variety F. The divisor diagram for F is \\begin{array}{l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 1 & 1 \\end{array}.
    \nNote that F is a scroll over \\CC P^1 with fibre \\CC P^3. There is a morphism F \\to \\CC P^4 , which is the blow up along x_0=x_1=0; this map sends [s_0,s_1,x, x_2, x_3, x_4] to [s_0x,s_1x,x_2,x_3,x_4]. There are two line bundles L, M on F: s_0,s_1 are section of L; xs_0, xs_1,x_2, x_3,x_4 are sections of M. X is cut out by a section of 3M: we have -K_X=L+M.Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(3m)!}{l! l! m! m! m! \\Gamma (1+m-l)}.
    \nRegularizing this gives period sequence 34:
    \n1 + 66t^2+816t^3+20214t^4+449640t^5+\\cdots.<\/li>\n
  6. a divisor of bidegree (2,2) in \\CC P^2 \\times \\CC P^2.\u00a0 Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(2l+2m)!}{l! l! l! m! m! m!}.
    \nRegularizing this gives period sequence 11:
    \n1+44 t^2+528 t^3+11292 t^4+228000 t^5+\\cdots.
    \nNote that this is a D3 form: even though the Fano X has rank 2, the quantum cohomology D-module splits off a “rank 1” irreducible piece (i.e. a piece of dimension 4, which is the size of the cohomology of a rank-1 Fano 3-fold).<\/li>\n
  7. the blow up of a quadric Q with centre the intersection of two members of \\cO (2). Thus, X is the complete intersection of two divisors in \\PP^1 \\times \\PP^4, of bidegrees (0,2) and (1,2). We denote by L and M the pull backs of the tautological bundles on the two factors.\u00a0 We have -K_X = L + M, and quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(2m)!(l+2m)!}{l! l! m! m! m! m! m!}.
    \nRegularizing this gives period sequence 51:
    \n1+36 t^2+348 t^3+6516 t^4+110880 t^5+\\cdots<\/li>\n
  8. a double cover of V_7 with branch locus a member B of |-K_{V_7}| such that B\\cap D is smooth, where D is the exceptional divisor of the blow-up V_7 \\to \\PP^3.\u00a0 This is a hypersurface in a toric variety F.\u00a0 The divisor diagram for F is \\begin{array}{llllll} 1 & 1 & 1 & -1 & 0 & 1 \\\\ 0 & 0 & 0 & 1 & 1 & 1 \\end{array}.\u00a0 Call the co-ordinates s_0, s_1, s_2, x, x_3, z.\u00a0 Let L be the line bundle with sections s_0,s_1,s_2 and let M be the line bundle with sections s_0 x, s_1 x, s_2 x, x_3; z is a section of L+M.\u00a0 The variety X is a section of 2L+2M on F; we have -K_X = L+M.Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(2l+2m)!}{l! l! l! \\Gamma(1+m-l) m! (l+m)!}.
    \nRegularizing this gives period sequence 26:
    \n1+26 t^2+216 t^3+3582 t^4+54480 t^5+\\cdots<\/li>\n
  9. the blow up of \\PP^3 in a curve \\Gamma of degree 7 and genus 5.\u00a0 \\Gamma is cut by the equations:
    \n\\rk \\left( \\begin{array}{lll} l_0 & l_1 & l_2 \\\\ q_0 & q_1 & q_2 \\end{array} \\right) < 2
    \nwhere the l_i are linear forms and the q_j are quadratics.\u00a0 Let y_0= l_0 q_1 - l_1 q_0, y_1 = l_2 q_0-l_0 q_2, y_2 = l_0 q_1 - l_1 q_0.\u00a0 The relations (szyzgies) between these equations are generated by:
    \n\\begin{cases} l_0 y_0 + l_1 y_1 + l_2 y_2 = 0 \\\\ q_0 y_0 + q_1 y_1 + q_2 y_2 = 0 \\end{cases}
    \nThus X is given by these two equations in \\PP^3 \\times \\PP^2, where the first factor has co-ordinates x_0, x_1, x_2, x_3 and the second factor has co-ordinates y_0, y_1, y_2.\u00a0 In other words, X is a complete intersection in \\PP^3 \\times \\PP^2 of type (1,1) \\cdot (2,1); we have -K_X = (1,1).
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+m)!(2l+m)!}{l! l!l!l! m! m! m!}.
    \nRegularizing this gives period sequence 62:
    \n1+22 t^2+174 t^3+2514 t^4+34200 t^5+\\cdots<\/li>\n
  10. the blow up of V_4 with centre an elliptic curve which is the intersection of two hyperplane sections.\u00a0 This is a complete intersection in a toric variety F. The divisor diagram for F is \\begin{array}{l l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 \\end{array}.
    \nNote that F is a scroll over \\PP^1 with fibre \\PP^4. There is a morphism F \\to \\PP^4, which is the blow up along x_0=x_1=0; this map sends [s_0,s_1,x, x_2, x_3, x_4,x_5] to [s_0x,s_1x,x_2,x_3,x_4,x_5]. There are two line bundles L, M on F: s_0,s_1 are section of L; xs_0, xs_1,x_2, x_3, x_4, x_5 are sections of M. X is a complete intersection of divisors 2M and 2M in F; we have -K_X=L+M. Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(2m)!(2m)!}{l! l! m! m! m! m!\\Gamma (1+m-l)}.
    \nRegularizing this gives period sequence 40:
    \n1+28 t^2+216 t^3+3516 t^4+49680 t^5+\\cdots<\/li>\n
  11. the blow up of V_3 with centre a line on it.\u00a0 This is a hypersurface in a toric variety F. The divisor diagram for F is \\begin{array}{llllll} 1 & 1 & 1 & -1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 1 & 1 \\end{array}.
    \nNote that F is a scroll over \\PP^2 with fibre \\PP^2. There is a morphism F \\to \\PP^4, which is the blow up along x_0=x_1=x_2=0; this map sends [s_0,s_1,s_2, x, x_3, x_4] to [s_0x,s_1x,s_2x,x_3,x_4]. There are two line bundles L, M on F: s_0,s_1 are section of L; xs_0, xs_1,xs_2, x_3, x_4 are sections of M. X is cut out by a section of L+2M in F; we have -K_X=L+M. Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+2m)!}{l! l! l! m! m!\\Gamma (1+m-l)}.
    \nRegularizing this gives period sequence 56:
    \n1+14 t^2+108 t^3+1074 t^4+13440 t^5+\\cdots<\/li>\n
  12. the blow up of \\PP^3 in a curve \\Gamma of degree 6 and genus 3.\u00a0 \\Gamma is cut by the equations:
    \n\\begin{pmatrix} l_{00} & l_{01} & l_{02} & l_{03} \\\\ l_{10} & l_{11} & l_{12} & l_{13} \\\\ l_{20} & l_{21} & l_{22} & l_{23} \\end{pmatrix}
    \nwhere the l_{ij} are linear forms.\u00a0 Let y_0,\\ldots,y_3 be the 3 \\times 3 minors.\u00a0 The relations (szyzgies) between these equations are generated by:
    \n\\begin{cases} l_{00} y_0 + l_{01} y_1 + l_{02} y_2 + l_{03} y_3 = 0 \\\\ l_{10} y_0 + l_{11} y_1 + l_{12} y_2 + l_{13} y_3 = 0 \\\\l_{20} y_0 + l_{21} y_1 + l_{22} y_2 + l_{23} y_3 = 0 \\end{cases}
    \nThus X is given by these three equations in \\PP^3 \\times \\PP^3, where the first factor has co-ordinates x_0, x_1, x_2, x_3 and the second factor has co-ordinates y_0, y_1, y_2,y_3.\u00a0 In other words, X is a complete intersection in \\PP^3 \\times \\PP^3 of type (1,1) \\cdot (1,1) \\cdot(1,1); we have -K_X = (1,1).
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+m)!(l+m)!(l+m)!}{l! l!l!l!m! m! m! m!}.
    \nRegularizing this gives period sequence 13:
    \n1+14 t^2+72 t^3+882 t^4+8400 t^5+\\cdots
    \nwhich is a D3 form (because this is obviously a G-Fano: it is Galkin’s Y_{20}).<\/li>\n
  13. the blow-up of a 3-dimensional quadric Q in a curve \\Gamma of genus 2 and degree 6.\u00a0 This is a complete intersection in a toric variety.\u00a0 We have \\Gamma = \\PP(1,1,3) \\cap Q where the embedding \\PP(1,1,3) \\hookrightarrow \\PP^4 sends [s_0:s_1:y] \\in \\PP(1,1,3) to [s_0^3:s_0^2 s_1:s_0 s_1^2:s_1^3:y] \\in \\PP^4.\u00a0 We\u00a0 blow up \\PP(1,1,3) inside \\PP^4.\u00a0 The equations defining \\PP(1,1,3) are
    \n\\rk \\begin{pmatrix} x_0 & x_1 & x_2 \\\\ x_1 & x_2 & x_3 \\end{pmatrix} < 2
    \nwhere [x_0:x_1:x_2:x_3:x_4] are co-ordinates on \\PP^4.\u00a0 The blown-up variety F is the complete intersection in \\PP^4 \\times \\PP^2 cut out by the equations:
    \n\\begin{cases} x_0 y_0 - x_1 y_1 + x_2 y_2 = 0 \\\\ x_1 y_0 - x_2 y_1 + x_3 y_2 = 0 \\end{cases}
    \nwhere y_0, y_1, y_2 are co-ordinates on \\PP^2.\u00a0 Our Fano X is the complete intersection of F with a quadric q(x_0,x_1,x_2,x_3,x_4).\u00a0 Thus X is a complete intersection of type (L+M)\\cdot(L+M)\\cdot(2L) in \\PP^4 \\times \\PP^2; here L is the tautological bundle on \\PP^4 and M is the tautological bundle on \\PP^2.
    \nWe have -K_X = L+N and Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+m)!(l+m)!(2l)!}{l! l!l!l!l! m! m! m!}.
    \nRegularizing this gives period sequence 52:
    \n1+14 t^2+84 t^3+930 t^4+9720 t^5+\\cdots<\/li>\n
  14. the blow-up…<\/li>\n
  15. the blow-up of \\PP^3 with center the intersection of a quadric and a cubic.\u00a0 This is a hypersurface in a scroll F.\u00a0 The divisor diagram for F is
    \n\\begin{array}{llllll} 1 & 1 & 1 & 1 & 0 & -1 \\\\ 0 & 0 & 0 & 0 & 1 & 1 \\end{array}.
    \nThe projection F \\to \\PP^3 sends [x_0,x_1,x_2, x_3, s, t] to [x_0,x_1,x_2,x_3]. There are two line bundles L, M on F: x_0,x_1,x_2,x_3 are section of L; s, tx_i are sections of M. X is cut out of F by a section of 2L+M; we have -K_X=L+M. Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(2l+m)!}{l! l! l! l! m!\\Gamma (1+m-l)}.
    \nRegularizing this gives period sequence 35:
    \n1+12 t^2+36 t^3+564 t^4+3600 t^5+\\cdots<\/li>\n
  16. the blow-up of V_4 \\subset \\PP^5 with center a conic on it.\u00a0 This is a complete intersection in a toric variety.\u00a0 We give full details of the construction, as it is a model for several other examples (being a blow-up of a projective hypersurface with center a complete intersection in the ambient projective space).We begin by constructing the blow-up Y of \\PP^5 with center the conic x_0=x_1=x_2=q=0 where q is a quadratic polynomial in x_0,\\ldots,x_5. \u00a0\u00a0 To do this, introduce new co-ordinates s_0, s_1, s_2, t, x and impose the relation:
    \n\\left(\\begin{array}{c} x_0 \\\\ x_1 \\\\ x_2 \\\\ q \\end{array}\\right) = x \\left(\\begin{array}{c} s_0 \\\\ s_1 \\\\ s_2 \\\\ t \\end{array}\\right).
    \nThus we construct Y as a hypersurface in the toric variety F with divisor diagram \\begin{array}{llllllll} 1 & 1 & 1 & -1 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\end{array}.\u00a0 The co-ordinates on F here are s_0,s_1,s_2,x,x_3,x_4,x_5,t; the equation defining Y is xt = q(s_0x,s_1x,s_2x,x_3,x_4,x_5).\u00a0 The map from Y to \\PP^5 sends [s_0,s_1,s_2,x,x_3,x_4,x_5] to [s_0 x, s_1 x, s_2 x, x_3, x_4, x_5].\u00a0 It is easy to check that this is the blow-up of \\PP^5 in the conic x_0=x_1=x_2=q=0.\u00a0 Introduce line bundles L, M such that s_0, s_1, s_2 are sections of L and s_0 x, s_1 x, s_2 x, x_3, x_4, x_5 are sections of M; note that Y is cut out of F by a section of 2M.\u00a0 The Fano X is a complete intersection of 3 divisors L+M, L+M, and 2M on F.\u00a0 We have -K_X = L+M.\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+m)!(l+m)!(2m)!}{l! l! l! \\Gamma (1+m-l) m! m! m! (l+m)!}.
    \nRegularizing this gives period sequence 59:
    \n1+10 t^2+60 t^3+510 t^4+4920 t^5+\\cdots<\/li>\n
  17. the blow-up…<\/li>\n
  18. the double cover of \\PP^1 \\times \\PP^2 with branch locus a divisor of bidegree (2,2).\u00a0 This is\u00a0 a hypersurface in a toric variety F with divisor diagram \\begin{array}{l l l l l l} 1 & 1 & 0 & 0 & 0 & 1\\\\ 0 & 0 & 1 & 1 & 1 & 1 \\end{array}. With coordinates x_0, x_1,y_0,y_1,y_2 ,z, the defining equation of X is z^2=f_{2,2}(x_0,x_1;y_0,y_1,y_2). Denote by L the line bundle with sections x_i and by M the line bundle with sections y_j; then -K_X = L+2M.Quantum Lefschetz gives
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{(2l+2m)!}{l! l! m! m! m! (l+m)!}.
    \nRegularizing this gives period sequence 60:
    \n1+6 t^2+48 t^3+282 t^4+2400 t^5+\\cdots<\/li>\n
  19. the blow-up of V_4 with center a line on it.\u00a0 We proceed as in example 16;\u00a0 X here is a complete intersection in the toric variety F with divisor diagram
    \n\\begin{array}{lllllll} 1 & 1 & 1 & 1 & -1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\end{array}.
    \nThe co-ordinates on F here are s_0,s_1,s_2,s_3,x,x_4,x_5,t; the map from Y to \\PP^5 sends [s_0,s_1,s_2,s_3,x,x_4,x_5] to [s_0 x, s_1 x, s_2 x, s_3 x, x_4, x_5].\u00a0\u00a0\u00a0 Introduce line bundles L, M such that s_0, s_1, s_2, s_3 are sections of L and s_0 x, s_1 x, s_2 x, s_3 x, x_4, x_5 are sections of M.\u00a0 Then X is a complete intersection of 2 divisors L+M, L+M on F.\u00a0 We have -K_X = L+M.\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+m}\\frac{(l+m)!(l+m)!}{l! l! l! l! \\Gamma (1+m-l) m! m!}.
    \nRegularizing this gives period sequence 55:
    \n1+8 t^2+30 t^3+240 t^4+1920 t^5+\\cdots<\/li>\n
  20. the blow-up…<\/li>\n
  21. the blow-up…<\/li>\n
  22. the blow-up…<\/li>\n
  23. the blow-up of a quadric with center an intersection of A \\in |\\cO(1)| and B \\in |\\cO(2)|.\u00a0 X is a complete intersection of type (L+2M)\\cdot(2M) in the toric variety with weight data \\begin{array}{llllllll} x_0 & x_1 & x_2 & x_3 & x_4 & s & t\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & L \\\\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & M \\end{array}.\u00a0 We have -K_X = L+2M.Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{(l+2m)!(2m)!}{m! m! m! m! m! l! (l+m)!}.
    \nRegularizing this gives period sequence 29:
    \n1+8 t^2+12 t^3+216 t^4+720 t^5+\\cdots<\/li>\n
  24. A divisor of bidegree (1,2) on \\PP^2 \\times \\PP^2.\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{2l+m}\\frac{(l+2m)!}{m! m! m! l! l! l!}.
    \nRegularizing this gives period sequence 66:
    \n1+4 t^2+24 t^3+132 t^4+780 t^5+\\cdots<\/li>\n
  25. The blow up of \\PP^3 with centre an elliptic curve which is the complete intersection of two quadrics. X is a divisor of bidegree (1,2) in \\PP^1 \\times \\PP^3. Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{(l+2m)!}{l! l! m! m! m! m!}.
    \nRegularizing this gives period sequence 28:
    \n1+4 t^2+24 t^3+60 t^4+720 t^5+\\cdots<\/li>\n
  26. the blow up…<\/li>\n
  27. the blow up of \\PP^3 with center a twisted cubic.\u00a0 The twisted cubic in \\PP^3 with co-ordinates x_0, x_1, x_2, x_3 is given by the condition
    \n\\rk \\left( \\begin{array}{lll} x_0 & x_1 & x_2 \\\\ x_1 & x_2 & x_3\\end{array} \\right) < 2.
    \nLet q_0= x_1 x_3 - x_2^2, q_1 = x_1 x_2-x_0 x_3, q_2 = x_0 x_2 - x_1^2.\u00a0 The relations (szyzgies) between these equations are generated by:
    \n\\begin{cases} x_0 q_0 + x_1 q_1 + x_2 q_2 = 0 \\\\ x_1 q_0 + x_2 q_1 + x_3 q_2 = 0 \\end{cases}
    \nThus X is given by these two equations in \\PP^3 \\times \\PP^2, where the first factor has co-ordinates x_0, x_1, x_2, x_3 and the second factor has co-ordinates q_0, q_1, q_2.\u00a0 In other words, X is a complete intersection in \\PP^3 \\times \\PP^2 of type (1,1) \\cdot (1,1).
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{2l+m}\\frac{(l+m)!(l+m)!}{l! l!l!l! m! m! m!}.
    \nRegularizing this gives period sequence 61:
    \n1+2 t^2+18 t^3+30 t^4+240 t^5+\\cdots<\/li>\n
  28. the blow-up of \\PP^3 with centre a plane cubic.\u00a0 X is a hypersurface of type (L+3M) in the toric variety with weight data
    \n\\begin{array}{lllllll} x_0 & x_1 & x_2 & x_3 & s & t\\\\ 0 & 0 & 0 & 0 & 1 & 1 & L \\\\ 1 & 1 & 1 & 1 & 0 & 2 & M \\end{array}.
    \nWe have -K_X = L+3M. Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+3m}\\frac{(l+3m)!}{m! m! m! m! l! (l+2m)!}.
    \nRegularizing this gives period sequence 33:
    \n1+18 t^3+24 t^4 + 0t^5 +\\cdots<\/li>\n
  29. the blow-up of a quadric 3-fold Q with centre a conic on it.\u00a0 X is a hypersurface of type 2M in the toric variety with weight data
    \n\\begin{array}{lllllll} s_0 & s_1 & x& x_2 & x_3 & x_4\\\\ 1 & 1 & -1 & 0 & 0 & 0 & L \\\\ 0 & 0 & 1 & 1 & 1 & 1 & M \\end{array}.
    \nWe have -K_X = L+2M. Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{(2m)!}{l! l! \\Gamma(1+m-l) m! m! m!}.
    \nRegularizing this gives period sequence 42:
    \n1+4 t^2+12 t^3+36 t^4+360 t^5+ \\cdots<\/li>\n
  30. the blow-up of \\PP^3 with center a conic.\u00a0 X is a hypersurface of type L+M in the toric variety with weight data
    \n\\begin{array}{lllllll} x_0 & x_1 & x_2 & x_3 & s & t\\\\ 0 & 0 & 0 & 0 & 1 & 1 & L \\\\ 1 & 1 & 1 & 1 & -1 & 0 & M \\end{array}.
    \nWe have -K_X = L+2M. Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{(l+m)!}{m! m! m! m! \\Gamma(1+l-m) l!}.
    \nRegularizing this gives period sequence 70:
    \n1+12 t^3+24 t^4 + 0 t^5+ \\cdots<\/li>\n
  31. the blow-up of a quadric 3-fold Q with center a line on it.\u00a0 X is a hypersurface of type L+M in the toric variety with weight data
    \n\\begin{array}{lllllll} s_0 & s_1 & s_2 & x & x_3 & x_4\\\\ 1 & 1 & 1 & -1 & 0 & 0 & L \\\\ 0 & 0 & 0 & 1 & 1 & 1 & M \\end{array}.
    \nWe have -K_X = L+2M. Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{(l+m)!}{l! l! l! \\Gamma(1+m-l) m! m!}.
    \nRegularizing this gives period sequence 48:
    \n1+2 t^2+12 t^3+6 t^4+180 t^5+ \\cdots<\/li>\n
  32. a divisor on \\PP^2 \\times \\PP^2 of bidegree (1,1).\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m\\geq 0} t^{2l+2m}\\frac{(l+m)!}{l! l! l! m! m! m!}.
    \nRegularizing this gives period sequence 6:
    \n1+4 t^2+60 t^4 + 0 t^5+ \\cdots
    \nNote that this is a D3 form.<\/li>\n
  33. the blow-up of \\PP^3 with center a line. X is a toric variety with weight data:
    \n\\begin{array}{llllll} s_0 & s_1 & x & x_2 & x_3 \\\\ 1 & 1 &- 1 & 0 & 0 & L \\\\ 0 & 0 & 1 & 1 & 1 & M \\end{array}.
    \nWe have -K_X = L+3M and:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+3m}\\frac{1}{l! l! \\Gamma(1+m-l) m! m!}.
    \nRegularizing this gives period sequence 54:
    \n1+6 t^3+24 t^4 + 0 t^5+ \\cdots<\/li>\n
  34. X = \\PP^1 \\times \\PP^2.\u00a0 We have:
    \nI_X = \\sum_{l,m\\geq 0} t^{2l+3m}\\frac{1}{l! l! m! m! m!}.
    \nRegularizing this gives period sequence 44:
    \n1+2 t^2+6 t^3+6 t^4+120 t^5 + \\cdots<\/li>\n
  35. V_7, which is the blow-up of \\PP^3 at a point.\u00a0 This is a toric variety with weight data:
    \n\\begin{array}{llllll} s_0 & s_1 & s_2 & x & x_3 \\\\ 1 & 1 & 1 & -1 & 0 & L \\\\ 0 & 0 & 0 & 1 & 1 & M \\end{array}.
    \nWe have -K_X = 2L+2M and:
    \nI_X = \\sum_{l,m\\geq 0} t^{2l+2m}\\frac{1}{l! l! l! \\Gamma(1+m-l) m!}.
    \nRegularizing this gives period sequence 30:
    \n1+2 t^2+30 t^4+ 0 t^5+ \\cdots<\/li>\n
  36. the scroll \\PP(\\cO \\oplus \\cO(2)) over \\PP^2.\u00a0 This is a toric variety with weight data:
    \n\\begin{array}{llllll} x_0 & x_1 & x_2 & s & t \\\\ 1 & 1 & 1 & -2 & 0 & L \\\\ 0 & 0 & 0 & 1 & 1 & M \\end{array}.
    \nWe have -K_X = L+2M and:
    \nI_X = \\sum_{l,m\\geq 0} t^{l+2m}\\frac{1}{l! l! l! \\Gamma(1+m-2l) m!}.
    \nRegularizing this gives period sequence 58:
    \n1+2 t^2+6 t^4+60 t^5+ \\cdots<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    We compute the quantum period sequences of Fano 3-folds in the Mori-Mukai list. [Not very Fano]The blow-up of with centre an elliptic curve which is the intersection of two members of\u00a0 . This is a hypersurface in a toric variety . \u00a0 The divisor diagram for is . Note that is a scroll over with […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":277,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-175","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/175","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=175"}],"version-history":[{"count":69,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/175\/revisions"}],"predecessor-version":[{"id":184,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/175\/revisions\/184"}],"up":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/277"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}