Let X be a del Pezzo surface of degree 3. By a classical result of Cayley-Salmon we know that X contains 27 lines counted with multiplicity. These can be used to construct a very affine variety X' and its tropicalization trop X' in the intrinsic torus. This embedding has the property that the 27 lines are "moved" outside the torus hence at the boundary of the tropical projective space. In [1] the authors show that trop X' can have two possible types. These are characterised by two arrangements of the 27 tropical lines at the boundary of trop X'.
From Question 2 we can see that we can also associate to every cubic V(f) in $\mathbb P^3$ six invariants. In order to obtain these we first need to compute the pentahedral form of X that is

$\{a_0x_0^3+a_1x_1^3+a_2x_2^3+a_3x_3^3+a_4x_4^3=0\} \cap \{ x_0+x_1+x_2+x_3+x_4 = 0\}$.


Can we identify cubics X=V( f ) that realize these two types by looking at the valuations of the six invariants in Question 2?

Tentative approach

The first thing to observe is that the result in [1] regards cubics embedded in the intrinsic torus while the six invariants are related to the embedding in $\mathbb P^3$ or more precisely in $\mathbb P^4$. However in [1] there is also a completely tropical construction of the tropicalization inside the intrinsic torus. This is obtained by a sequence of tropical modifications associated to 6 special points. They correspond to the 6 points P1,…,P6 of $\mathbb P^2$ that are blown up to obtain X(see [2] Theorem 3.2 and [3] Example 7.22). In particular, if we know that the points P1,…,P6 realise one of the generic types then Theorem 5.1 in [1] then we can deduce that the type of the tropicalization.

Group Leader

Sara Lamboglia


Marvin Hahn


[1] Ren, Q., Shaw, K., Sturmfels, B. . Tropicalization of del Pezzo surfaces.

[2] Mourtada, H., Sarıoğlu, C.C., Soulé, C., Zeytin, A. (Eds.). Algebraic Geometry and Number Theory

[3]Harris, J,. Algebraic Geometry: A First Course

[4] Q. Ren, S. Sam, B. Sturmfels. Tropicalization of Classical Moduli Spaces

[5] T.G. Berry, Richard R. Patterson. Implicitization and parametrization of nonsingular cubic surfaces

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