# Rational Simple Connectedness

# Rational Simple Connectedness

Rational simple connectedness is an analog of simple connectedness for complex varieties having important applications: every 2-parameter family of rationally simply connected varieties has a rational section, and a 1-parameter family has so many rational sections that they approximate every power series section to arbitrary order. Unfortunately the condition is quite difficult to verify and is known to hold only for homogeneous spaces and also for some projective hypersurfaces satisfying a list of hypotheses. My new approach for verifying this condition works by studying a canonically defined foliation on the moduli space of rational curves on the variety. By proving integrability of this foliation, I prove every smooth complete intersections $X$ of type $(d_1, \ldots, d_c)$ in $\mathbb{P}^n$ is rationally simply connected whenever $\sum d_i^2 \leq n$ and when the associated moduli space of lines on $X$ is smooth. This degree range is sharp.