What are the degree $6$ curves in a cubic surface $X$ other than complete intersection of $X$ with a quadric? Is there any such degree $6$ curve in X? If yes, then can it contain an elliptic curve of degree $4$?
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2$\begingroup$ Yes, a lot. For every $g$ with $0\leq g\leq 4$, there are many families of curves of degree $6$ and genus $g$. And of course you can have the union of an elliptic curve of degree 4 and of a conic, if this is the meaning of your last question. Please look at "cubic surfaces" on any textbook of algebraic geometry. $\endgroup$– abxCommented May 24, 2020 at 13:43
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$\begingroup$ why all such curves contained in a cubic surface ? why the last example in not in any quadric ? $\endgroup$– user130022Commented May 24, 2020 at 13:59
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2$\begingroup$ A cubic surface is obtained by blowing up 6 points in the plane. From this description it is easy to construct all examples you want. $\endgroup$– abxCommented May 24, 2020 at 14:28
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