On the number of complement components of hypersurface coamoebas

Transcription

1 On the number of complement components of hypersurface coamoebas Texas A&M University, College Station, AMS Sectional Meeting Special Sessions: On Toric Algebraic Geometry and Beyond, Akron, OH, October 20, Joint work with Frank Sottile

2 Summary

3 Preliminaries Let C := C \ {0} Log Re C Arg exp R Im C R π R/2πZ = S 1

4 Preliminaries Let V f be the complex algebraic hypersurface defined by the polynomial f (z) = a α z α, α supp(f ) with a α C, and supp(f ) finite subset of Z n V f = {z (C ) n f (z) = 0}

5 Preliminaries V f (C ) n Log Arg exp A Vf R n Re C n Im R n π (S 1 ) n coa Vf

6 Definition of Amoebas and Coamoebas DEFINITION The amoeba of V f is A Vf := Log(V f ) The coamoeba of V f is coa Vf := Arg(V f )

7 Phase limit set Let S(V ) be the set of sequences {z n } V such that the sequence {Log(z n )} is unbounded. Let q = {z n } be an element of S(V ), and acc(q) be the set of accumulation points, in the real torus (S 1 ) n, of the sequence {Arg(z n )}. Definition Let V (C ) n be an algebraic variety. The phase limit set of V is the subset of the real torus (S 1 ) n denoted by P (V ) and defined by : P (V ) := q S(V ) acc(q).

8 Phase Limit Sets Theorem (Nisse-Sottile) Let V be a complex algebraic hypersurface with defining polynomial f with its Newton polytope. Then, the phase limit set of V is the union of the closure of the hypersurface coamoebas with defining polynomials the truncations of f to the facets of, i.e., P (V ) = coa (V f Γ j ) Γ j facet of

9 Phase Limit Sets Lemma (Mikhalkin & Nisse-Sottile) Let C (C ) n be an algebraic curve, and coa its coamoeba. Then coa = coa P (C) where P (C) is an arrangement of circles.

10 Preliminaries Theorem (Nisse-Sottile) The phase limit set contains an arrangement of k-dimensional tori. In the case of hypersurfaces, this arrangement bounds a (k + 1)-chain which we call the shell of the coamoeba, and we denote by H (V ).

11 Phase limit set Figure: On the left we have the phase limit set of the complex plane curve defined by f θ, ρ (z, w) = ρe iθ z + w + z 3 w + z 2 w 2, on the right top the shell of its coamoeba.

12 Preliminaries Figure: The complex coamoeba which is equal to the Non-Archimedean coamoeba of the the complex plane curve defined by f θ, 1.

13 Main Theorem Theorem (Nisse-Sottile) Let V be a complex algebraic hypersurface defined by a polynomial f with Newton polytope. If n > 2, then we have the following inequalities hold : #{(S 1 ) n \ coa (V )} #{(S 1 ) n \ P (V )} #{(S 1 ) n \ P (V )} #{(S 1 ) n \ H (V )} #{(S 1 ) n \ H (V )} n! Vol( ), where #A denotes the number of connected components of the set A.

14 Main Theorem If n = 2, then the phase limit set is of dimension one, and the last inequalities become : #{(S 1 ) 2 \coa (V )} #{(S 1 ) 2 \H (V )} 2 Area( ). Moreover, we give an example of an integer polygon 0 R 2 where the last inequality is never sharp for any complex plane curve with such Newton polygon. Using this example, we obtain for any n an example of a polytope where the last inequality is never sharp.

15 Examples of coamoebas of some complex algebraic plane curves Figure: Coamoeba of some real curve

16 Examples of coamoebas of some complex algebraic plane curves Figure: Coamoeba of some complex curve

17 Questions Figure: The coamoeba of the line and its spine

18 Questions Figure: The quiver dual to the spine with one vertex v 1 and three edges a, b, and c. The oriented subdivision of the torus dual to the spine of the coamoeba in the latest figure. What is the analogue of this construction in higher dimension?

19 The End Thank you for your attention

20 The End Thank you for your attention