Tropical Fano schemes

Abstract

We define a tropical version Fd(tropX)$\operatorname{F}_d(\operatorname{trop}X)$ of the Fano Scheme Fd(X)$\operatorname{F}_d(X)$ of a projective variety X⊆Pn$X\subseteq \mathbb {P}^n$ and prove that Fd(tropX)$\operatorname{F}_d(\operatorname{trop}X)$ is the support of a polyhedral complex contained in prefixtropdouble-struckGfalse(d,nfalse)$\operatorname{trop}\mathbb {G}(d,n)$. In general, prefixtropFd(X)⊆Fd(tropX)$\operatorname{trop}\operatorname{F}_d(X)\subseteq \operatorname{F}_d(\operatorname{trop}X)$ but we construct linear spaces L$L$ such that prefixtropF1(X)⊊F1(tropX)$\operatorname{trop}\operatorname{F}_1(X)\subsetneq \operatorname{F}_1(\operatorname{trop}X)$ and show that for a toric variety prefixtropFd(X)=Fd(tropX)$\operatorname{trop}\operatorname{F}_d(X)=\operatorname{F}_d(\operatorname{trop}X)$.

Moduli Spaces of Codimension-One Subspaces in a Linear Variety and their Tropicalization

“…If X 1 and X 2 are two (d + 1)-dimensional linear subspaces tropicalizing to the same Bergman fan X , it is still possible that the tropicalization of the moduli spaces of ddimensional subspaces trop(L(X 1 )) and trop(L(X 2 )) differ (see, for instance, Example 25 below, and Examples 3.3 and 3.4 in [31]). However, trop(L(X)) is constant whenever X is a subspace in the generic set (see Proposition 11).…”

“…For an example of the generic subset of a Grassmannian, see Example 13 above. In the next example, we relate our computation in Example 13 of the space of generic planes for the Bergman fan of U 3,6 to Lamboglia's Fano schemes [31].…”

“…Let X be the Bergman fan of U 3,6 . In [31], Lamboglia considers the space of all (realizable) tropical lines contained in X , and which is a 3-dimensional polyhedral complex that strictly contains the tropicalized spaces of lines that we study here. She also provides concrete examples to show the space trop(L(X)) depends on the choice of the lift X of X : two different linear subspaces X 1 and X 2 tropicalizing to X are presented, and the tropicalizations of their spaces of lines L(X 1 ) and L(X 2 ) are computed.…”

“…She also provides concrete examples to show the space trop(L(X)) depends on the choice of the lift X of X : two different linear subspaces X 1 and X 2 tropicalizing to X are presented, and the tropicalizations of their spaces of lines L(X 1 ) and L(X 2 ) are computed. The linear subspace X 1 is the rowspace of the matrix [31]). This subspace is generic in our sense: the relations given in Example 13 are all nonzero (where we use the dehomogenized version involving q 123 = 1).…”

“…Thus, self-duality does not necessarily hold for tropical planes. This observation motivates us to ask the following question (also studied in [31]): Question 2. Given a tropicalized plane X ⊂ R n−1 , what is the space of tropicalized lines in X ?…”