Certificate Del Pezzo

The Certificate for del Pezzo surfaces of degree 3 contains a certificate which invalidates Conjecture 5.3 of

[RSS16] Q. Ren, K. Shaw, B. Sturmfels: Tropicalization of Del Pezzo Surfaces, Adv. Math. 300 (2016), 156–189. arXiv:1402.5651

It works in the polynomial ring

\[\mathbb Q(d_1,\dots,d_6)[E_1,\dots,E_6,F_{12},F_{13},\dots,F_{56},G_1,\dots,G_6]\]

in which \(d_i\) represent generic complex numbers determining generic points \(p_i:=(1:d_i:d_i^3)\in\mathbb P_{\mathbb C}^2\) and for the smooth cubic surface \(X:=\text{Bl}_{p_1,\ldots,p_6}(\mathbb P_{\mathbb C}^2)\):

  • \(E_i\) represents the exceptional divisor over the point \(p_i\),

  • \(F_{ij}\) represents the strict transform of the line through \(p_i\) and \(p_j\),

  • \(G_i\) represents the strict transform of the conic through \(\{p_1,\ldots,p_6\} \setminus \{p_i\}\).

The script contains a generating set \(F\) for the universal Cox Ideal \(I_X\) up to saturation as specified in Proposition 2.2 of [RSS16]. For a specific weight vector \(w_{\text{max}}\in\mathbb R^{\{E_i, F_{ij}, G_i\}}\), the script shows

  • \(w_{\text{max}}\in\text{Trop}(F)\) by verifying that all initial forms \(\text{in}_{w_{\text{max}}}(f)\) for \(f\in F\) have at least 2 terms

  • \(w_{\text{max}}\notin\text{Trop}(I_X)\) by verifying that the initial ideal \(\text{in}_{w_{\text{max}}}(I_X)\) contains the monomial \(E_6F_{56}G_6\).

                    SINGULAR                              /  Development
A Computer Algebra System for Polynomial Computations    /   version 4.1.1
by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann  \   Feb 2018
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
> LIB "tropicalBasis.lib"; // initializes necessary libraries and helper functions
> intvec wMin = 2,1,0,1,1,1,0,2,0,0,0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0;
                                 // wMin is in min-convention
> intvec wMax = -wMin;           // Singular uses max-convention
> intvec allOnes = onesVector(size(wMax));
> ring r = (0,d1,d2,d3,d4,d5,d6),(E1,E2,E3,E4,E5,E6,
.   F12,F13,F14,F15,F16,F23,F24,F25,F26,F34,F35,F36,F45,F46,F56,
.   G1,G2,G3,G4,G5,G6),(a(allOnes),a(wMax),lp);
                          // prepending allOnes makes no difference mathematically
                          // as the ideal is homogeneous,
                          // but it helps computationally
> ideal F =               // Singular ideals are list of polynomials
.  (d3-d4)*(d1+d3+d4)*E2*F12+(d2-d4)*(d1+d2+d4)*E3*F13-(d2-d3)*(d1+d2+d3)*E4*F14,
   [...]                  // see uploaded file
.  -(d5-d6)*(d1+d3+d4)*F24*G4+(d4-d6)*(d1+d3+d5)*F25*G5-(d4-d5)*(d1+d3+d6)*F26*G6;
> ideal inF = initial(F,wMax); // initial forms of the elements in F
                               // all are at least binomial, hence wMax lies in Trop(F)
> ideal IX = groebner(F);
> ideal inIX = initial(IX,wMax);// initial forms of Groebner basis elements
                                // this is a Groebner basis of in_wMax(IX)
> reduce(E6*F56*G6,inIX);       // normal form is 0 hence E6*F56*G6 lies in in_wMax(IX)