Math Art
Enriques Quintic, 4 Tacnodes (Craighero-Gattazzo (1:1) member)

Enriques Quintic, 4 Tacnodes (Craighero-Gattazzo (1:1) member)

Enriques Quintic, 4 Tacnodes (Craighero-Gattazzo (1:1) member) is an algebraic surface, defined by an implicit equation, immersed with singularities.

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algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

(x2⁢(y+z)+y2⁢(x+z)+z2⁢(x+y)+x⁢y⁢z+(x⁢y+x⁢z+y⁢z)+(x+y))⁢(x⁢y+x⁢z+y⁢z)+(z−1)2⁢(x+y)=0\left(x^{2} \left(y + z\right) + y^{2} \left(x + z\right) + z^{2} \left(x + y\right) + x y z + \left(x y + x z + y z\right) + \left(x + y\right)\right) \left(x y + x z + y z\right) + \left(z - 1\right)^{2} \left(x + y\right) = 0

Properties

Family
algebraic
Given by
an implicit equation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed with singularities
Fidelity
exact
Exactness
elementary

Definition

One of Stagnaro's Enriques quintics: a degree-5 surface with 4 tacnodes, in a family whose parameters set the tacnode type (0, 1 or 2). THE SHIPPED ROW IS THE CRAIGHERO-GATTAZZO SECTION-3 PENCIL MEMBER (lambda : mu) = (1 : 1) -- the open-access completion of Stagnaro's program, same singular configuration, irreducible bicanonical adjoint -- NOT Stagnaro's own [St.1] surface, whose equation remains unpublished outside LNM 997. One typo in the source found by testing its claims: the text's fourth Dd-point P(1,0,0,1) must read (0,0,1,1); with that the four Dd-points, the cone tangencies and the rank-1 tacnodal Hessians all verify at machine precision (gated).

Sources

Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 0).