137x Filetype PDF File size 0.44 MB Source: projecteuclid.org
Rev. Mat. Iberoamericana 27(2011), no.1, 181 232 Tropical plane geometric constructions: a transfer technique in Tropical Geometry Luis Felipe Tabera Abstract The notion of geometric construction is introduced. This notion allows to compare incidence congurations both lying in the algebraic andthetropical plane. We providesufficient conditions in a geometric construction to ensure that there is always an algebraic counterpart related by tropicalization. We also present some results to detect if this algebraic counterpart cannot exist. With these tools, geometric constructions are applied to transfer classical theorems to the tropical framework, we provide a notion of constructible incidence theorem and then several tropical versions of classical theorems are proved such as the converse of Pascals, Fanos or Cayley-Bacharach theo- rems. 1. Introduction Let K be an algebraically closed eld provided with a nontrivial rank one valuation v and valuation group T. We suppose that T is a subgroup of the ∗ reals, v : K → T ⊆ R. We have the following map on the algebraic torus: ∗ n n T :(K) → T (x ,...,x ) → (v(x ),...,v(x )) 1 n 1 n This mapis the tropicalization or projection map. Tropical varieties are then ∗ n dened as the image of an algebraic variety V ⊆ (K ) under the tropicaliza- tion map T. One of the most interesting aspects of tropical varieties is that they inherit relevant geometric properties from their algebraic counterparts. In the present work we explore this inheritance for the case of translating 2000 Mathematics Subject Classification: Primary: 14H99, Secondary: 51E30, 12J25, 16Y60. Keywords: Tropical geometry, geometric constructions, incidence congurations. 182 L.F. Tabera incidence theorems of classical projective geometry to the tropical context. The origin of this work is the Pappus theorem counterexample in [11]. In that paper, a tropical conguration of points and lines in the shape of Pap- pus theorem hypotheses is shown such that it does not verify Pappus thesis. In particular, it implies that this conguration is not the projection of a similar conguration of points and lines in the algebraic plane. The authors provided then another alternative version of the same theorem and claimed that this new version would hold in the tropical context. The key of this newversion of Pappus theorem is that the hypotheses are given as the result of a geometric construction dealing with points and lines. The correctness of this theorem was shown in [13] using some precursor techniques on geo- metric constructions. Following this idea, many incidence theorems can be given as a construction of a conguration of curves and points (hypothesis) and then some information is derived (the thesis of the theorem). Thus, we will focus on geometric constructions in the plane and how they behave with respect to tropicalization. Intuitively, a geometric construction is a procedure that starts with a set of input curves and points and then denes other curves and points by either intersecting two available curves or computing a curve dened by a polynomial of xed support passing through a set of points (a conic through ve points, for example). The main algorithm we present consists in tak- ing a tropical instance of a geometric construction and then computing a constructible set S, over the residual eld of the valuation, that encodes sufficient conditions for the compatibility with tropicalization of an alge- braic geometric construction. We will also show some certicates during the computation to detect if a tropical realization of a geometric construction is not the projection of any algebraic realization. Moreover, we present a notion of admissible geometric construction.This is a combinatorial notion that ensures that for all tropical realizations of the construction, the computed set S isnonemptyanddense.Thatis,there will always be an algebraic preimage of the construction under the tropical- ization T. This notion can be applied to prove that some incidence theorems (so-called constructible incidence theorems) hold in the tropical context if we are able to describe their hypotheses as the output of an admissible geometric construction. The paper is structured as follows: in Section 2 we present the notion of geometric construction and show how to understand the steps of a con- struction in both the algebraic and the tropical context. In Section 3 we provide the main algorithm of the article. Then, the limits of our geometric construction method are shown by a series of examples. Furthermore, we include a generalization of the notion of admissibility related with the notion Tropical Plane Geometric Constructions 183 of points in general position with respect to a curve. Finally, in Section 4, we use the results obtained so far to build up a notion of constructible incidence theorem that is compatible with tropicalization and we show some relevant instances of theorems of this kind. 1.1. Notation and preliminaries Let k be the residual eld of K by the valuation. There are three main cases of valued elds according to the characteristics: the case char(K)= char(k) = 0 (equicharacteristic zero), the case char(K)=char(k)=p>0 (positive characteristic) and the case char(K)=0
no reviews yet
Please Login to review.