{"id":5153,"date":"2010-12-03T14:19:27","date_gmt":"2010-12-03T14:19:27","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5153"},"modified":"2011-02-17T11:44:20","modified_gmt":"2011-02-17T11:44:20","slug":"things-are-not-as-straightforward-as-they-seem","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5153","title":{"rendered":"Things are not as straightforward as they seem"},"content":{"rendered":"

Consider the blow-up X of \\PP^1 \\times \\PP^1 with center a complete intersection of type (2,1)\\cdot(1,1).\u00a0 Since the complete intersection consists of three points, X is a del Pezzo surface dP_5.\u00a0 It is tempting to compute its regularized period sequence as follows.<\/p>\n

Warning: this calculation is wrong.<\/strong> I explain below where the error is and how to fix it.\u00a0 We express X as a complete intersection in a toric variety F as follows.\u00a0 Let F have weight data:
\n\\begin{array}{ccccccc} x_0 & x_1 & y_0 & y_1 & s & t & \\\\ 1 & 1 & 0 & 0 & 0 & -1 & L \\\\ 0 & 0 & 1 & 1 & 0 & 0 & M \\\\ 0 & 0 & 0 & 0 & 1 & 1 & N \\end{array}
\nNow consider the equation:
\ns f_{1,1} + t g_{2,1} = 0
\nwhere f_{1,1} and g_{2,1} are polynomials in x_i, y_j of bidegrees (respectively) (1,1) and (2,1).\u00a0 The variety X defined by this equation is cut out by a section of the line bundle L+M+N; by projecting [x_0:x_1:y_0:y_1:s:t] \\mapsto [x_0:x_1:y_0:y_1] we see that X is, as desired, the blow-up of \\PP^1 \\times \\PP^1 in a complete intersection of type (2,1)\\cdot(1,1).\u00a0 We have -K_X = M+N and:
\nI_X(t) = \\sum_{l,m,n \\geq 0} t^{m+n} {(l+m+n)! \\over l!l!m!m!n!(n-l)!}
\nRegularizing gives the period sequence:
\nI_{reg}(t) = 1-3 t+23 t^2-105 t^3+783 t^4-4053 t^5+29729 t^6+\\cdots<\/p>\n

This is not correct<\/strong>: we know the regularized period sequences<\/a> for del Pezzo surfaces, and in the case of dP_5 we get:
\nI_{reg}(t) =1+12 t^2+42 t^3+468 t^4+3360 t^5+31350 t^6 + \\cdots
\nSo what went wrong?<\/p>\n

The construction of X given above is correct.\u00a0 It is the second half of the calculation which is flawed.\u00a0 The key point is that, even though X is Fano and is cut out of the ambient space F by a section of an ample line bundle, -K_X is not the restriction of an ample line bundle on F but rather is only the restriction of a semi-positive<\/em> line bundle on F.\u00a0 Thus the mirror map is non-trivial.<\/em> To see this we need to consider not Golyshev’s I-function:
\nI_X(q) = \\sum_{d} q^{-K_X\\cdot d} {\\prod_{i} (E_i\\cdot d)! \\over \\prod_j (D_j \\cdot d)!}
\nbut rather the full Givental I-function:
\nI^{Giv}_X(q) = q_1^{D_1\/z}\\cdots q_r^{D_r\/z} \\sum_{d} q_1^{d_1}\\cdots q_r^{d_r} \\prod_{i} {\\Gamma(1+E_i\/z+E_i\\cdot d) \\over \\Gamma(1+E_i\/z)} \\prod_j {\\Gamma(1+D_j\/z) \\over \\prod_j \\Gamma(1+D_j\/z+D_j \\cdot d)} z^{K_X \\cdot d}
\nHere X is cut out of the toric variety with toric divisors D_j by a section of the direct sum of line bundles \\oplus_i E_i.\u00a0 Golyshev’s I-function is obtained from Givental’s I-function by taking the term in cohomological degree zero and setting:
\n\\begin{cases} q_1 = q^{k_1} \\\\ \\vdots \\\\ q_r = q^{k_r} \\\\ z = 1 \\end{cases}
\nwhere -K_X = k_1 D_1 + \\ldots + k_r D_r.\u00a0 Note that Givental’s I-function is homogeneous of degree zero if we set \\deg q_i = k_i, \\deg z = 1, and \\deg \\alpha = m whenever \\alpha \\in H^{2m}(X).<\/p>\n

In the situation at hand (i.e. X is a semipositive complete intersection in a toric variety) we have, for grading reasons:
\nI^{Giv}_X(q) = q_1^{D_1\/z}\\cdots q_r^{D_r\/z} \\Big(F(q) + G(q)\/z + H_1(q) D_1\/z + \\cdots + H_r(q) D_r\/z + O(z^{-2}) \\Big)
\nwhere F, H_1,\\ldots,H_r are degree-zero power series in the q_i and G is a degree-1 power series in the q_i.\u00a0 Furthermore Givental’s mirror theorem states that:
\n{exp\\Big(-{G(q) \\over z F(q)}\\Big) \\over F(q)} I^{Giv}_X(q) = J_X(\\hat{q})
\nwhere:
\n\\begin{cases} \\hat{q}_1 = q_1 \\exp(H_1(q)\/F(q)) \\\\ \\vdots \\\\ \\hat{q}_r = q_r \\exp(H_r(q)\/F(q)) \\end{cases}
\nThis change of variables is called the mirror map<\/em>.\u00a0 The regularized quantum period sequence that we seek is obtained from the cohomological-degree-zero\u00a0 component of the J-function by setting \\hat{q}_i = t^{k_i}, z=1, and doing the trick with factorials: \\sum_k a_k t^k \\longmapsto \\sum_k k! a_k t^k.<\/p>\n

Applying this discussion in our case (X = dP_5 realized as above) yields:
\n\\begin{cases} F(q) = 1 \\\\ G(q) = q_2+q_3+2q_1 q_3 \\\\ H_1(q) = \\sum_{k>0} {(-1)^{k} \\over k} q_1^k = {-\\log(1+q_1)} \\\\ H_2(q) = 0 \\\\ H_3(q) = \\log(1+q_1) \\end{cases}
\nand hence:
\n\\begin{cases} q_1 = {\\hat{q}_1 \\over 1 - \\hat{q}_1} \\\\ q_2 = \\hat{q_2} \\\\ q_3 = \\hat{q}_3 (1-\\hat{q_1}) \\end{cases}
\nThus the cohomological-degree-zero part of the J-function is:
\n\\exp(-\\hat{q}_2-\\hat{q}_3(1-\\hat{q}_1)+2\\hat{q}_1 \\hat{q}_3) \\sum_{k,l,m\\geq 0} \\hat{q}_1^k \\hat{q}_2^l \\hat{q}_3^m(1-\\hat{q}_1)^{m-k} {1 \\over z^{l+m}} {(k+l+m)! \\over k!k!l!l!m!(m-k)!}
\nand setting \\hat{q}_0 = 1, \\hat{q}_1 = t, \\hat{q}_2 = t, z=1 yields:
\n\\exp(-3t) \\sum_{k,l\\geq 0} t^{k+l} {(2k+l)! \\over k!k!l!l!k!}
\nRegularizing this gives:
\nI_{reg}(t) =1+12 t^2+42 t^3+468 t^4+3360 t^5+31350 t^6 + \\cdots
\nwhich agrees with our previous calculation.<\/p>\n","protected":false},"excerpt":{"rendered":"

Consider the blow-up of with center a complete intersection of type .\u00a0 Since the complete intersection consists of three points, is a del Pezzo surface .\u00a0 It is tempting to compute its regularized period sequence as follows. Warning: this calculation is wrong. I explain below where the error is and how to fix it.\u00a0 We […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-5153","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5153","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/types\/post"}],"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=5153"}],"version-history":[{"count":32,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5153\/revisions"}],"predecessor-version":[{"id":5365,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5153\/revisions\/5365"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5153"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5153"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}