In algebraic geometry, a Segre surface, studied by and , is an intersection of two

quadrics In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...

in 4-dimensional projective space. They are

rational surface In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of su ...

s isomorphic to a

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

blown up in 5 points with no 3 on a line, and are

del Pezzo surface In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general ...

s of degree 4, and have 16 rational lines. The term "Segre surface" is also occasionally used for various other surfaces, such as a quadric in 3-dimensional projective space, or the hypersurface :

x_1 x_2 x_3 + x_2 x_3 x_4 + x_3 x_4 x_5 + x_4 x_5 x_1 + x_5 x_1 x_2 = 0. \,

References

* *{{Citation , doi=10.1093/qmath/2.1.216 , last1=Segre , first1=Beniamino , title=On the inflexional curve of an algebraic surface in S₄ , mr=0044861 , year=1951 , journal=The Quarterly Journal of Mathematics , series=Second Series , issn=0033-5606 , volume=2 , issue=1 , pages=216–220 Algebraic surfaces Complex surfaces