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<head><title>Conics on a quintic threefold -- Example using Schubert2</title>
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<div><h1>Conics on a quintic threefold -- Example using Schubert2</h1>
<div><div>The number of conics (=rational curves of degree 2) on a general quintic hypersurface in P4 was computed by S. Katz in 1985. Here is how the computation can be made with Schubert2.</div>
<div>Any conic in P4 spans a unique plane. Hence the space of conics in P4 is a certain P5-bundle over the Grassmannian <tt>G</tt> of planes in P4. This space is called <tt>X</tt> in the following code:</div>
<table class="examples"><tr><td><pre>i1 : G = flagBundle({2,3}) -- Grassmannian of planes in P4
o1 = G
o1 : a flag bundle with ranks {2, 3}</pre>
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<tr><td><pre>i2 : (S,Q) = G.Bundles -- Q = rank 3 tautological quotient bundle
o2 = (S, Q)
o2 : Sequence</pre>
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<tr><td><pre>i3 : B = symmetricPower(2,Q) -- The bundle of quadratic forms on the variable plane
o3 = B
o3 : an abstract sheaf of rank 6 on G</pre>
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<div>As a matter of convention, a projectiveBundle in Schubert2 parametrizes the rank 1 quotients. The P5-bundle of conics is given by sublinebundles of B, so we take the dual in the following:</div>
<table class="examples"><tr><td><pre>i4 : X = projectiveBundle(dual B, VariableNames => {,{z}})
o4 = X
o4 : a flag bundle with ranks {5, 1}</pre>
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<div>The equation of the general quintic is a section of the fifth symmetric power of the space of linear forms on P4. The induced equation on any given conic is an element in the corresponding closed fiber of a certain vector bundle <tt>A</tt> of rank 11 on the parameter space <tt>X</tt>. On any given plane P, and conic C in P, we get the following exact sequence: <p align=center><i> 0 →H<sup>0</sup>(O<sub>P</sub>(3)) →H<sup>0</sup>(O<sub>P</sub>(5)) →H<sup>0</sup>(O<sub>C</sub>(5)) →0</i></p> As C varies, these sequences glue to a short exact sequence of bundles on X, to give</div>
<table class="examples"><tr><td><pre>i5 : A = symmetricPower_5 Q - symmetricPower_3 Q ** OO(-z)
o5 = A
o5 : an abstract sheaf of rank 11 on X</pre>
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<div>A given conic is contained in the quintic if and only if the equation of the quintic vanishes identically on the conic. Hence the class of the locus of conics contained in the quintic is the top Chern class of <tt>A</tt>. Hence their number is the integral of this Chern class:</div>
<table class="examples"><tr><td><pre>i6 : integral chern A
o6 = 609250
o6 : QQ[]</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Lines_spon_sphypersurfaces.html" title="Example using Schubert2">Lines on hypersurfaces</a> -- Example using Schubert2</span></li>
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