A geometric version of the circle method. (English) Zbl 1493.14047

Let \(X\subset\mathbb{A}^n\) be a smooth hypersurface defined by a polynomial \(f\) of degree \(k\ge 3\) over a field \(K\), and assume that the leading terms of \(f\) define a smooth projective hypersurface, \(Z\) say. For a fixed \(K\)-point \(P=[(a_1,\ldots,a_n)]\) of \(Z\) one is interested in the space Mor\(_{d,P}(\mathbb{A}^1,X)\) of \(n\)-tuples of polynomials \((g_1,\ldots,g_n)\) of degree \(d\), with leading terms \(a_1,\ldots,a_n\), satisfying \(f(g_1,\ldots,g_n)=0\). Then it is shown that Mor\(_{d,P}(\mathbb{A}^1,X)\) is irreducible and has the expected dimension \(d(n-k)\) for any field \(K\) whose characteristic is either zero or \(>k\), provided that \(d\ge k-1\ge 2\) and \[ \left\lfloor\frac{d}{k-1}\right\rfloor \left(\frac{n}{2^k}-k+1\right)\ge 1. \] This is proved as a corollary to the main theorem, which describes the compactly supported cohomology of the space of rational curves on a smooth hypersurface.
To establish the principal result the authors use “spreading out”, so that it suffices to examine the situation over the algebraic closure of a finite field. The key new feature of the proof is a geometric analogue of the circle method. The major arcs are handled geometrically, although the treatment is guided by calculations familiar from the traditional setting. For the minor arcs the problem reduces to a point counting problem for function fields over finite fields, where existing circle method techniques apply.


14H10 Families, moduli of curves (algebraic)
11P55 Applications of the Hardy-Littlewood method
14F20 Étale and other Grothendieck topologies and (co)homologies
14G05 Rational points
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