Notes on Real Rationally Connected Varieties and Fano Threefolds of Genus 12

We show that a smooth projective geometrically rationally connected variety over the real numbers with at least one rational point admits a non-constant mapping from a smooth projective curve. Additionally, we show that many real smooth Fano complete intersections admit non-constant maps from the real anisotropic conic. Furthermore, we compute the genus and degree of the singular locus of the locus of lines on a genus 12 Fano threefold. After blowing up this locus to obtain simple normal crossings divisor, we compute the cohomology of the complement, in which we see the genus of this curve appear in weight 5 of the third cohomology group