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Published online by Cambridge University Press: 17 March 2022
In this chapter we discuss advanced tools and techniques, which rely on additional concepts from algebraic geometry. These tools could be helpful for people who do research work in incidence theory and related topics. A reader who is new to this field might prefer to skip this chapter.
We sometimes wish to consider families of varieties, such as the set of circles in the plane or the set of planes in R^3 that not are incident to the origin. In this chapter, we rigorously define such families. We also generalize the idea of point-line duality to every family of varieties. We then see how these notions could be used to prove various results. In particular, we derive a new incidence bound and prove various properties of surfaces in R^3 and C^3.
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