Quantitative Arithmetic of Projective Varieties

Manin's conjecture for certain biprojective hypersurfaces

“…We note that the degree of M y is bounded independently of y. Thus, the proof of Theorem 3.1 in [6] delivers…”

Manin’s conjecture for quartic Del Pezzo surfaces with a conic fibration

“…where now L k1,k2 (B) is the number of (s, t) ∈ Λ(k, λ, 1) such that (1)-(3) and (6) hold in the definition of L(B). We now consider the regions (5.6) and (5.7) in more detail.…”

“…Here K is given by (6.6) and L k1,k2,ℓ (B) denotes the number of (s, t) ∈ Λ(k, λ, ℓ) such that (2), (3) and (6) hold in the definition of L(B), with 2 ∤ gcd(s, t) and ν 2 (s) 2Z 2 . Note that it will facilitate our ensuing investigation to restrict the summation to odd values of ℓ here.…”

“…In the present situation, with the quadratic form (1.4), the fibres that we need to examine have the shape 6) for coprime a, b ∈ Z. It is clear that C a,b ⊂ P 2 Q is a non-singular plane conic when the discriminant ∆(a, b) = −2(a 4 − b 4 ) is non-zero.…”

Equivariant Compactifications of Two-Dimensional Algebraic Groups

“…See [Bro09,Chapter 2] for an overview. In the version stated in [BT98b], Manin's conjecture is expected to hold for any del Pezzo surface whose singularities are rational double points (i.e.…”