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

## Synopsis

• Function: euler
• Usage:
euler V
• Inputs:
• Outputs:
• an integer, the topological Euler characteristics of the variety V

## Description

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

## Caveat

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