Tropical projective space

Conventional visualization of the tropical projective plane, with projection of the real coordinate axes.

In tropical geometry, a tropical projective space is the tropical analog of the classic projective space.

Definition

Given a module Template:Math over the tropical semiring Template:Math, its projectivization is the usual projective space of a module: the quotient space of the module (omitting the additive identity Template:Math) under scalar multiplication, omitting multiplication by the scalar additive identity 0:Template:Efn

$𝐓(M):=(M\setminus \mathrm{𝟎})/(𝐓\setminus 0).$

In the tropical setting, tropical multiplication is classical addition, with unit real number 0 (not 1); tropical addition is minimum or maximum (depending on convention), with unit extended real number Template:Math (not 0),Template:Efn so it is clearer to write this using the extended real numbers, rather than the abstract algebraic units:

$𝐓(M):=(M\setminus \mathrm{\infty })/(𝐓\setminus \mathrm{\infty }).$

Just as in the classical case, the standard Template:Mvar-dimensional tropical projective space is defined as the quotient of the standard Template:Math-dimensional coordinate space by scalar multiplication, with all operations defined coordinate-wise:Template:Sfn

${𝐓𝐏}^n:=({𝐓}^{n+1}\setminus \mathrm{\infty })/(𝐓\setminus \mathrm{\infty }).$

Tropical multiplication corresponds to classical addition, so tropical scalar multiplication by Template:Math corresponds to adding Template:Math to all coordinates. Thus two elements of Template:Tmath are identified if their coordinates differ by the same additive amount Template:Math:

$(x_0,\dots ,x_n)\sim (y_0,\dots ,y_n)⟺(x_0+c,\dots ,x_n+c)=(y_0,\dots ,y_n).$

References

• Template:Cite arXiv
• Template:Cite arXiv