Vol 83, No 4 (2019)

Let $\mathcal{C} \subset [0,1]$ be a set satisfying the descending chaincondition. We show that every accumulation point of volumes of log canonicalsurfaces $(X, B)$ with coefficients in $ \mathcal{C} $ can be realized asthe volume of a log canonical surface with big and nef $K_X+B$ and withcoefficients in $\overline{\mathcal{C}} \cup \{1 \}$ in such a way that atleast one coefficient lies in $\operatorname{Acc} (\mathcal{C}) \cup \{1 \}$.As a corollary, if $\overline {\mathcal{C}} \subset \mathbb{Q}$, then allaccumulation points of volumes are rational numbers. This proves a conjectureof Blache. For the set of standard coefficients$\mathcal{C}_2=\{1-1/{n} \mid n\in\mathbb{N} \} \cup \{1 \}$ we prove thatthe minimal accumulation point is between $1/{(7^2 \cdot 42^2)}$ and$1/{42^2}$.

We study the bounded derived categories of torus-equivariant coherent sheaveson smooth toric varieties and Deligne–Mumford stacks. We construct anddescribe full exceptional collections in these categories. We also observethat these categories depend only on the $\mathrm{PL}$-homeomorphism typeof the corresponding simplicial complex.

We prove that a Fano complete intersection of codimension $k$ and index $1$in the complex projective space ${\mathbb P}^{M+k}$ for $k\ge 20$ and$M\ge 8k\log k$ with at most multi-quadratic singularities is birationallysuperrigid. The codimension of the complement of the set of birationallysuperrigid complete intersections in the natural moduli space is shown tobe at least $(M-5k)(M-6k)/2$. The proof is based on the technique ofhypertangent divisors combined with the recently discovered$4n^2$-inequality for complete intersection singularities.

This paper contains geometric tools developed to solve the finite-fieldcase of the Grothendieck–Serre conjecture in [1]. It turns out thatthe same machinery can be applied to solve some cohomological questions.In particular, for any presheaf of $S^1$-spectra $E$ on the category of$k$-smooth schemes, all its Nisnevich sheaves of $\mathbf{A}^1$-stablehomotopy groups are strictly homotopy invariant. This shows that $E$ is$\mathbf{A}^1$-local if and only if all its Nisnevich sheaves of ordinarystable homotopy groups are strictly homotopy invariant. The latter resultwas obtained by Morel [2] in the case when the field $k$ is infinite.However, when $k$ is finite, Morel's proof does not work since it usesGabber's presentation lemma and there is no published proof of that lemma.We do not use Gabber's presentation lemma. Instead, we develop the machineryof nice triples invented in [3]. This machinery is inspired byVoevodsky's technique of standard triples [4].

We classify smooth Fano threefolds with infinite automorphism groups.