Chern classes of two natural vector bundles over the grassmannian . There is a sequence of vector bundleswhere is the trivial vector bundle of rank , the fiber of over is the subspace , and is the quotient vector bundle (which exists since the rank is constant on each of the fibers). The Chern classes of these two associated bundles arewhere is an -tuple andThe tautological sequence then gives the presentation of the Chow ring as G(2,4) One of the classical examples analyzed is the Grassmannian since it parameterizes lines in . Schubert calculus can be used to find the number of lines on a Cubic surface. Chow ring The Chow ring has the presentation and as a graded Abelian group it is given by Lines on a cubic surface This Chow ring can be used to compute the number of lines on a cubic surface. Recall a line in gives a dimension two subspace of , hence . Also, the equation of a line can be given as a section of . Since a cubic surface is given as a generic homogeneous cubic polynomial, this is given as a generic section . Then, a line is a subvariety of if and only if the section vanishes on . Therefore, the Euler class of can be integrated over to get the number of points where the generic section vanishes on . In order to get the Euler class, the total Chern class of must be computed, which is given asThen, the splitting formula reads as the formal equationwhere and for formal line bundles . The splitting equation gives the relations and .Since can be read as the direct sum of formal vector bundleswhose total Chern class ishenceusing the fact and Then, the integral issince is the top class. Therefore there are lines on a cubic surface. See also Enumerative geometry Chow ring Intersection theory Grassmannian Giambelli's formula Pieri's formula Chern class Quintic threefold Mirror symmetry conjecture References Summer school notes http://homepages.math.uic.edu/~coskun/poland.html Phillip Griffiths and Joseph Harris (1978), Principles of Algebraic Geometry, Chapter 1.5 David Eisenbud and Joseph Harris (2016), "3264 and All That: A Second Course in Algebraic Geometry". Algebraic geometry Topology of homogeneous spaces