Let I be a toric ideal. We say I is robust if its universal Gröbner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graphtheoretic conditions on the set of circuits of G.Theorem 1.2. I G is robust if and only if the following conditions are satisfied.R1: No circuit of G has an even chord, R2: No circuit of G has a bridge, R3: No circuit of G contains an effective crossing, and R4: No circuit of G shares exactly one edge (and no other vertices) with another circuit such that the shared edge is part of a cyclic block in both circuits.

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We introduce higher-order variants of the Frobenius-Seshadri constant due to Mustaţȃ and Schwede, which are defined for ample line bundles in positive characteristic. These constants are used to show that Demailly's criterion for separation of higher-order jets by adjoint bundles also holds in positive characteristic. As an application, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg. We also discuss connections with other characterizations of projective space.

In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs (Y, ∆) over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when Y surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when Y is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective nonvanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.

In this expository paper, we show that the Deligne-Mumford moduli space of stable curves is projective over Spec(Z). The proof we present is due to Kollár. Ampleness of a line bundle is deduced from nefness of a related vector bundle via the ampleness lemma, a classifying map construction. The main positivity result concerns the pushforward of relative dualizing sheaves on families of stable curves over a smooth projective curve.