We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and of the special fiber ring generate the same ideal in the symmetric algebra. We show that the ideal defining the special fiber ring is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible height. An important step in the proof is the calculation of the degree of the variety parametrized by the forms generating the height three Gorenstein ideal.

We develop an explicit “higher-rank” Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of number fields. We show this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture and, as a first application of this approach, we prove new cases of the conjecture over natural families of abelian CM-extensions of totally real fields for which the relevant $p$-adic $L$-functions possess trivial zeroes.

A. Venkatesh raised the following question, in the context of torsion automorphic forms: can the mod $p$ analogue of Grothendieck’s standard conjecture of Künneth type be true (especially for compact Shimura varieties)? In the first theorem of this article, by using a topological obstruction involving Bockstein, we show that the answer is in the negative and exhibit various counterexamples, including compact Shimura varieties.

It remains an open geometric question whether the conjecture can fail for varieties with torsion-free integral cohomology. Turning to the case of abelian varieties, we give upper bounds (in $p$) for possible failures, using endomorphisms, the Hodge–Lefschetz operators, and invariant theory.

The Schottky problem enters into consideration, and we find that, for the Jacobians of curves, the question of Venkatesh has an affirmative answer for every prime number $p$.

We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor ${\mathcal{C}}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that ${\mathcal{C}}_{42}$ is uniruled.

We present a quantitative version of Bilu’s theorem on the limit distribution of Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus. Our result gives, for a given point, an explicit bound for the discrepancy between its Galois orbit and the uniform distribution on the compact subtorus, in terms of the height and the generalized degree of the point.

We establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.

Let $R$ be a commutative noetherian ring. Denote by ${\mathsf{D}}^{-}\left(R\right)$ the derived category of cochain complexes $X$ of finitely generated $R$-modules with ${\mathsf{H}}^i\left(X\right)=0$ for $i\gg 0$. Then ${\mathsf{D}}^{-}\left(R\right)$ has the structure of a tensor triangulated category with tensor product $\cdot {\otimes }_R^L\cdot$ and unit object $R$. In this paper, we study thick tensor ideals of ${\mathsf{D}}^{-}\left(R\right)$, i.e., thick subcategories closed under the tensor action by each object in ${\mathsf{D}}^{-}\left(R\right)$, and investigate the Balmer spectrum $\mathtt{Spc}{\mathsf{D}}^{-}\left(R\right)$ of ${\mathsf{D}}^{-}\left(R\right)$, i.e., the set of prime thick tensor ideals of ${\mathsf{D}}^{-}\left(R\right)$. First, we give a complete classification of the thick tensor ideals of ${\mathsf{D}}^{-}\left(R\right)$ generated by bounded complexes, establishing a generalized version of the Hopkins–Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum $\mathtt{Spc}{\mathsf{D}}^{-}\left(R\right)$ and the Zariski spectrum $SpecR$, and study their topological properties. After that, we compare several classes of thick tensor ideals of ${\mathsf{D}}^{-}\left(R\right)$, relating them to specialization-closed subsets of $SpecR$ and Thomason subsets of $\mathtt{Spc}{\mathsf{D}}^{-}\left(R\right)$, and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of ${\mathsf{D}}^{-}\left(R\right)$ in the case where $R$ is a discrete valuation ring.