The first Hirzebruch surface (the blow-up of $\mathbb{P}^2$ at one point) is a projective toric surface that naturally embeds into $\mathbb{P}^4$ as a cubic surface (sometimes called the cubic scroll). As a real algebraic variety, its real part is homeomorphic to a Klein bottle. I am confused about the designation "cubic surface": I (naively) thought that a generic cubic surface in $\mathbb{P}^4$ would be the intersection of a cubic hypersurface with a hyperplane. But one cannot embed the Klein bottle in a hyperplane of $\mathbb{RP}^4$ (i.e. in $\mathbb{RP}^3$). What is the (probably quite dumb) misconception on my part ?
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3$\begingroup$ "Cubic surface" here just means of degree 3. It is definitely not contained in a hyperplane. $\endgroup$– abxCommented Nov 26, 2023 at 16:11
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2$\begingroup$ To elaborate on abx's comment, degree three means that a general 2-plane intersects it in 3 points. There are multiple types of cubic surface in 4-space. $\endgroup$– Phil TostesonCommented Nov 26, 2023 at 16:15
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$\begingroup$ @PhilTosteson , abx, thank you for your comment. There is no relation between the degree of the surface and the degree of the defining equations of the surface then ? $\endgroup$– YromedCommented Nov 26, 2023 at 16:24
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1$\begingroup$ There is a relation if a variety is a complete intersection, which is not the case for the cubic scroll. $\endgroup$– SashaCommented Nov 26, 2023 at 16:44
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