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V. Drinfeld and E. Lau introduced a ``decompletion'' of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors,…

We bound the dimension of the Prym-Brill-Noether variety for an open subset of the moduli space of \'{e}tale double covers of k-elliptic curves. We also obtain new bounds on the dimension of the Prym-Brill-Noether variety for general…

We introduce the intersection theory of the moduli space of curves and its tautological ring. We survey open questions about the tautological ring and sketch techniques for proving the Chow ring is or is not generated by tautological…

Jin and Rubinstein asked whether the fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold, and proved this equality for toric varieties. In this paper we provide a positive answer…

It was recently established by Perepechko and Zaidenberg that the automorphism group of a normal affine surface is finite-dimensional if and only if the surface admits no non-trivial action of the additive group of the base field. We extend…

A Nikulin surface is the minimal resolution of the quotient of a $K3$ surface $S$ by a symplectic involution $\iota_S$. Equivalently, it is the $2$-dimensional component of the fixed locus of the involution induced by $\iota_S$ on the…

Motivated by applications of algebraic geometry to reconstruction problems in computer vision, we initiate a study of the equations of degeneracy loci associated with linearly dependent points on Segre varieties. When these points are…

Let $(X,\Theta)$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class $\theta_k=[\Theta]^k/k!$ for $k<g$. We show that the minimal positive multiple of $\theta_k$ which is…

We give an overview of the theory of quasi-$F$-singularities, focusing on their connection with singularities in birational geometry.

We study arrangements of smooth conics and lines in the complex projective plane whose singularities are limited to nodes, tacnodes, and ordinary triple points. The first part of the paper gives numerical restrictions for plus-one generated…

We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature $(2, n)$ with $n\equiv 1, 3$ mod $8$ and discriminant $-2$ in the range…

Let $X$ be a Gorenstein minimal $3$-fold of general type whose canonical map is generically finite. We prove that if $p_g(X)> 243$, then the degree of the canonical map is at most $72$. Moreover, equality holds only if the general fibre $F$…

The moduli space of meromorphic Higgs bundles admits a Poisson structure due to the independent work of Bottacin and Markman. In this paper, we revisit the symplectic leaves of this Poisson structure for the tame case. We study the partial…

We study Witt-vector affine Springer fibers for tame equi-valued conjugacy classes in tamely ramified groups. Similar to the approach of Goresky-Kottwitz-MacPherson in the equal characteristic setting, we show that they admit pavings by…

We define log conifold transitions for Fano threefold pairs of index two and study their deformation theory. Relying on the recent solution to the relative Clemens conjectures in this setting, we construct rational curves with normal bundle…

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations without the assumption on the nef part in the complex analytic setting. We also record the corresponding algebraic statement.

We continue the study of Pascal-type residual constructions in projective four-space. Starting from two $k$-tuples of hyperplanes in $\mathbb P^4$ such that the $k$ diagonal intersection planes are contained in a hyperplane, one obtains a…

Classical nonabelian Hodge theory identifies Dolbeault and de Rham moduli spaces by providing a real-analytic isomorphism. In this paper, motivated by the Kapustin--Witten theory, we study this correspondence in the more general framework…

Let $X$ be a Picard-rank-one del Pezzo manifold of dimension $n\geq 4$ over an algebraically closed field of characteristic zero. Okamura proved that the unpointed Kontsevich spaces $\overline{M}_{0,0}(X,d)$ are irreducible of the expected…

Farkas, Pandharipande, and Sammartano constructed non-rational irreducible components of Hilbert schemes of points in affine space $\mathbb{A}^n$ for all $n \geq 12$. Their construction starts from Hilbert schemes of curves in…