EulerCharacteristic -- topological Euler characteristic of a (smooth) projective variety

Synopsis

• Usage:

  EulerCharacteristic I

• Inputs:
  • I, an ideal, a homogeneous ideal defining a smooth projective variety $X\subset\mathbb{P}^n$
• Optional inputs:
  • BlowUpStrategy => ..., default value Eliminate,
  • MathMode => ..., default value false, whether to ensure correctness of output
  • Verbose => ..., default value true,
• Outputs:
  • an integer, the topological Euler characteristics of $X$.

Description

This is an application of the method SegreClass. See also the corresponding methods in the packages CSM-A, by P. Aluffi, and CharacteristicClasses, by M. Helmer and C. Jost.

In general, even if the input ideal defines a singular variety $X$, the returned value equals the degree of the component of dimension 0 of the Chern-Fulton class of $X$. The Euler characteristic of a singular variety can be computed via the method ChernSchwartzMacPherson.

In the example below, we compute the Euler characteristic of $\mathbb{G}(1,4)\subset\mathbb{P}^{9}$, using both a probabilistic and a non-probabilistic approach.

i1 : I = Grassmannian(1,4,CoefficientRing=>ZZ/190181);

                ZZ
o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
              190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4

i2 : time EulerCharacteristic I
     -- used 0.36301 seconds

o2 = 10

i3 : time EulerCharacteristic(I,MathMode=>true)
MathMode: output certified!
     -- used 0.0971544 seconds

o3 = 10

Caveat

See also

• euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
• ChernSchwartzMacPherson -- Chern-Schwartz-MacPherson class of a projective scheme
• SegreClass -- Segre class of a closed subscheme of a projective variety