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These notes give a short introduction to the study of curves on algebraic varieties. The main emphasis is on families of genus 0 curves. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected varieties. The last section is devoted to a conjectural characterization of rationally connected varieties using symplectic geometry.
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János Kollár, Holomorphic and pseudo-holomorphic curves on rationally connected varieties. Port. Math. 67 (2010), no. 2, pp. 155–179DOI 10.4171/PM/1863