TROPICAL SCHEME THEORY

Transcription

1 TROPICAL SCHEME THEORY 1. Tropical ideals The idea of tropical scheme theor in general is the following. An ideal I K[ 1,..., n ] gives rise to a variet V (I) A n K a variet. With classical tropicalization we get a polhedral comple trop(v ) R n, where R = (R { }, = min, = +). Scheme theoretic tropicalization remembers more information. For eample, the data of B(I), the bend relations, are equivalent to remembering trop(i) = trop(f) f I (ou need ideal generated b if the valuation isn t surjective). So we get some ideals in R[ 1,..., n ], but which ideals in R[ 1,..., n ] do we like for tropical geometr? It seems we don t want to consider all of the ideals. Some complications: (1) R[ 1,..., n ] is not Noetherian. For eample, trop( ) =, 2 2, 2 2,... is not finitel generated, and so the chain of ideals given b taking the first n of these gives an infinite ascending chain of ideals in R[, ]. (2) If I R[ 1,..., n ] we define V (I) := { R n the minimum in f() is achieved twice f I}. This gives too much. Proposition 1.1. An conve closed subset S of R n whose affine hull has dimension at most n 1 (i.e. which is contained in some affine hperplane) and such that aff(s) is a rational subspace is of the form V (I) for some ideal I. Proof. Note that an such S is an intersection of rational hperplanes and rational half-hperplanes. An rational hperplane is the bend locus of a binomial. And we can get an rational half-hperplane b intersecting two tropical varieties which are shifts of each other. So we restrict which ideals we look at. Definition 1.2. An ideal I R[ 1,..., n ] is a tropical ideal if I d := {f I deg f d} R Mons d (where Mons d is the set of monomials of degree d) is a tropical linear space for ever d. Equivalentl, I is a tropical linear space if for an f, g I with some monomial u of degree at most d such that [f] u = [g] u then there is some h I such that [h] u = (tropical ) and for an monomial v, [h] v min([f] v, [g] v) with equalit if [f] v [g] v. Date: October 26, 217, Speaker: Felipe Rincón, Scribe: Netanel Friedenberg. 1

2 2 TROPICAL SCHEME THEORY This is a proposed definition to solve some of the above problems. u C Eample 1.3. If I = trop(j) for J K[ 1,..., n ] then J is a tropical ideal. In this case we sa that I is realizable. I = is not a tropical ideal: In degree 2 we have 2 and 2, but this would force 2 2 I. Warning: Tropical ideals are generall not finitel generated. (There are trivial eceptions, such as monomial ideals.) There are non-realizable tropical ideals. Take I R[, ] generated (as a semi-module) b polnomials of the form f = u for C a minimal collection of k + 1 monomials in Mons d inside a standard triangle of size k. For eample, in degree 1 we have which is a standard triangle of size 2, so I. In degree 2 we have 2 2 From we see that 2 I, and from

3 TROPICAL SCHEME THEORY 3 we see that 2 2 I. Note that in degree is not in I because we alread have 2 2 in I. It is not hard to see that this gives a tropical ideal. (What we have done is described the circuits of the matroid.) V (I) = V ( ). But I is not realizable: If I = trop(j) then sa J so ( + + 1)( ) = J but trop( ) is not in I because it has too few monomials. (Note that up to scaling each of the variables b a scalar with valuation, which doesn t change the tropicalizations, we will will have J, so this is the onl case we need to consider.) Nice things about tropical ideals:

4 4 TROPICAL SCHEME THEORY If I is a tropical ideal then we get a Hilbert function H I : Z Z H I (d) = dim(i d ) - while dimension is not well-behaved for arbitrar semi-modules, it is wellbehaved for tropical linear spaces. If I = trop(j) then H I = H J. Proposition 1.4. H I (d) is eventuall polnomial for an tropical ideal. Theorem 1.5. There is no infinite ascending chain of tropical ideals. I 1 I 2 I 3 Theorem 1.6. If I is a tropical ideal then V (I) is (the support of) a finite polhedral comple. Let I be a tropical ideal. We have the bend congruence B(I) = f fû f I, u supp(f). Then we can recover I as I = {f R[ 1,..., n ] B(f) B(I)}. The main point is that the proof of this for the classical case (i.e. tropicalized ideals) reall onl took advantage of dualit for valuated matroids. Let s talk about wh the Hilbert function is eventuall polnomial and wh there are no infinite ascending chains of tropical ideals. The use the ideal of initial degenerations. Consider a weight vector w R n (think of as assigning a weight to each variable). For f R[ 1,..., n ] write f = u N n a u u, then in w f := a u w u =f(w) u B[ 1,..., n ]. If I R[ 1,..., n ] is an ideal then in w (I) := in w (f) f I. Ke fact: If I is a tropical ideal then in w (I) is also a tropical ideal. The bases of M d (in w (I)) are the bases B of M(I d ) such that p d (B) i B w i is minimal. In the triviall valued case this is taking the corresponding face of the matroid poltope. In the non-triviall valued case this goes b taking the correct piece of the corresponding regular subdivision of the matroid poltope. In particular, because the bases don t change size, this gives us that H inw(i) = H I. Proof that H I (d) is eventuall polnomial. If I is a tropical ideal then for generic w R n in w (I) is a monomial ideal. Because monomial ideals are the same classicall and tropicall, H inw(i) is eventuall polnomial, H I = H inw I is eventuall polnomial. Proof that tropical ideals satisf the ACC. Take an infinite chain I 1 I 2 I 3

5 TROPICAL SCHEME THEORY 5 of tropical ideals. For ver generic w R n (we just have to avoid a countable collection of codimension-1 things) in w (I 1 ) in w (I 2 ) in w (I 3 ) is a chain of monomial ideals. But chains of monomial ideals must stabilize. So H In = H inw(i n) must stabilize. But if I J are tropical ideals with the same Hilbert function then we must have I = J. This follows from a basic matroidal fact if L 1 L 2 are tropical linear spaces of the same dimension then L 1 = L 2.