In this paper, we describe a new infinite family of q 2 −1 2 -tight sets in the hyperbolic quadrics Q + (5, q), for q ≡ 5 or 9 mod 12. Under the Klein correspondence, these correspond to Cameron-Liebler line classes of PG(3, q) having parameter q 2 −1 2 . This is the second known infinite family of nontrivial Cameron-Liebler line classes, the first family having been described by Bruen and Drudge with parameter q 2 +1 2 in PG(3, q) for all odd q.The study of Cameron-Liebler line classes is closely related to the study of symmetric tactical decompositions of PG(3, q) (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when q ≡ 9 mod 12 (so q = 3 2e for some positive integer e), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler in 1982; the nature of these decompositions allows us to also prove the existence of a set of type 1 2 (3 2e − 3 e ), 1 2 (3 2e + 3 e ) in the affine plane AG(2, 3 2e ) for all positive integers e. This proves a conjecture made by Rodgers in his PhD thesis.

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the linear strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology, and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface ν 6 (P 2 ) ⊆ P 27 in characteristic 40 009. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d (P 2 ).

Abstract. Let R be a number field or a recursive subring of a number field and consider the polynomial ring R [T ]. We show that the set of polynomials with integer coefficients is diophantine over R [T ]. Applying a result by Denef, this implies that every recursively enumerable subset of R[T ] k is diophantine over R [T ].

We construct a Diophantine interpretation of F q [W, Z] over F q [Z]. Using this together with a previous result that every recursively enumerable (r.e.) relation over F q [Z] is Diophantine over F q [W, Z], we will prove that every r.e. relation over F q [Z] is Diophantine over F q [Z]. We will also look at recursive infinite base fields F, algebraic over F p . It turns out that the Diophantine relations over F [Z] are exactly the relations which are r.e. for every recursive presentation.

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