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euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety



The command computes the topological Euler characteristic of the (smooth) projective variety V as an alternated sum of its Hodge numbers. The Hodge numbers can be computed directly using the command hh.

A smooth plane quartic curve has genus 3 and topological Euler characteristic -4:
i1 : Quartic = Proj(QQ[x_0..x_2]/ideal(x_0^4+x_1^4+x_2^4))

o1 = Quartic

o1 : ProjectiveVariety
i2 : euler(Quartic)

o2 = -4

The topological Euler characteristic of a smooth quintic hypersurface in projective fourspace is -200:
i3 : Quintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-101*x_0*x_1*x_2*x_3*x_4))

o3 = Quintic

o3 : ProjectiveVariety
i4 : euler(Quintic)

o4 = -200


No test is made to see if the projective variety is smooth

See also

Ways to use this method: