We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold (P 1 , D), where. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms.
On the leading constant in the Manin-type conjecture for Campana points by Alec Shute (Klosterneuburg)1. Introduction. The study of Campana points is an emerging area of interest in arithmetic geometry as a way to interpolate between rational and integral points. Campana orbifolds, first introduced in [4] and [5], consist of a variety X and a weighted boundary divisor D of X. The Campana points associated to the orbifold (X, D) can be viewed as rational points of X that are integral with respect to D. In the recent paper [16], Pieropan, Smeets, Tanimoto and Várilly-Alvarado formulate a Manin-type conjecture for the quantitative study of Campana points on Fano Campana orbifolds, which henceforth we shall refer to as the PSTV-A conjecture. The authors establish their conjecture in the special case of vector group compactifications, using the height zeta function method developed by Chambert-Loir and Tschinkel [7], [8].The arithmetic study of Campana points is still in its early stages. Initial results in [2], [22] and [3], which predate the formulation of the PSTV-A conjecture, concern squareful and m-full values of hyperplanes of P n+1 . (We recall that a non-zero integer z is m-full if for any prime p dividing z, we have p m | z, and squareful if it is 2-full.) Following discussions in the Spring 2006 MSRI program on rational and integral points on higherdimensional varieties, Poonen [17] posed the problem of finding the number of coprime integers z 0 , z 1 such that z 0 , z 1 and z 0 + z 1 are all squareful and bounded by B. In the language of the PSTV-A conjecture, this corresponds to counting Campana points on the orbifold (P 1 , D), where D is the divisorUpper and lower bounds for this problem were obtained by Browning and Van Valckenborgh [2], but finding an asymptotic formula remains wide open. Van Valckenborgh [22] considers a higher-dimensional
A central question in Arithmetic geometry is to determine for which polynomials f ∈ Z[t] and which number fields K the Hasse principle holds for the affine equation f (t) = N K/Q (x) = 0. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.
In [3], Poonen and Slavov develop a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.
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