Degenerations of complete collineations and geometric Tevelev degrees of ℙ𝑟

We consider the problem of enumerating maps 𝑓 of degree 𝑑 from a fixed general curve 𝐶 of genus 𝑔 to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> </m:math> Pʳ satisfying incidence conditions of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>p</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>X</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:math> f(pᵢ)∈ Xᵢ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mi>C</m:mi> </m:mrow> </m:math> pᵢ∈ C are general points and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> </m:mrow> </m:math> Xᵢʳ are general linear spaces. We give a complete answer in the case where the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>X</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> Xᵢ are points, where the counts are known as the “Tevelev degrees” of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mi>r</m:mi> </m:msup> </m:math> Pʳ . These were previously known only when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> r=1 , when 𝑑 is large compared to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> </m:math> r,g , or virtually in Gromov–Witten theory. We also give a complete answer in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> r=2 with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations.