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We prove a conjecture of da Silva Machado and Seade that characterizes weighted homogeneous isolated hypersurface singularities through the existence of a logarithmic vector field transverse to the link. For a reduced isolated hypersurface…

Recently in [6] the authors proposed a conjecture that the homogeneity of an isolated hypersurface germ can be detected by the existence of non-degenerate holomorphic logarithmic vector fields. In this paper we prove this conjecture…

We study the positivity properties of finite flat quotients of a normal projective variety. The numerical groups and the positive cones of these quotient varieties are related to those of the original variety.

In this expository article, we present on state-of-the art results regarding three closely related invariants of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields. We study the moduli space…

Given a 3-term perfect complex E over a quasi-projective variety X and a nonnegative integer r, we define two virtual cycles and their refinements supported over the r-th degeneracy loci of E. This is done by modifying the complex E after…

We give examples of varieties $X$ defined over a non-algebraically closed field $k$ with nontrivial unramified cohomology, in the case when the field $k$ is of bounded cohomological dimension, or the variety $X$ is a conic bundle over a…

Let \(S_b\) be the class of birational morphisms between smooth varieties over a field \(F\), and let \(L_n=S_b^{-1}d_{\leq n}\Sm(F)\). Kahn and Sujatha asked whether the natural functor \(L_n\to S_b^{-1}\Sm(F)\) is fully faithful. We prove…

We provide an intrinsic characterization of hyperelliptic stable curves of genus $g \geq 2$, independent of admissible covers or auxiliary moduli data. A stable curve is hyperelliptic if it admits an involution yielding a rational tree…

In this paper, we prove a Dolbeault geometric Langlands equivalence for $\GL_r$ and for the Langlands dual pair $\SL_r/\PGL_r$ over an open locus of the Hitchin base which strictly contains the elliptic locus. This open locus contains the…

Let $X$ be a smooth irreducible projective variety of dimension $n\ge 3$ over an algebraically closed field of characteristic zero, polarized by a very ample line bundle $\OO_X(1)$. Let $\E$ be an Ulrich bundle on $X$. We prove that there…

Let $B$ be a curve on an irrational ruled surface $X$. We prove that the logarithmic Kodaira dimension of $X-B$ equals the Iitaka dimension of $K_X+B$ and give a rough configuration of $B$ when the logarithmic Kodaira dimension of $X - B$…

Slowness surfaces are algebraic varieties arising from propagation of elastic waves. In dimensions $2$, we completely classify the types of singularities slowness surfaces can have. The two types of possible singularities are a transversal…

The blowing-up of the projective plane at a finite set of points yields a del Pezzo surface if and only if the points lie in general position. In this note, we generalize this result to Severi--Brauer surfaces over arbitrary ground fields.…

We study the maximal variation problem for linear systems associated with a very ample line bundle, using Hodge theory and Picard-Lefschetz theory. We provide an affirmative answer to the maximal variation problem for a broad class of…

In this paper, we study pencils of plane curves of sufficiently large degree $d$ with simple base points, and their reducible curves whose irreducible components have degree at most $k\geq 2$. Combining techniques from algebraic geometry…

We study smooth polarized projective varieties $(X,H)$ whose exterior powers of the tangent bundle are Ulrich. We prove that if $\bigwedge^rT_X$ is $H$-Ulrich for some $0<r<\dim X$, then $X$ is Fano and the intersection number $(-K_X)\cdot…

We construct higher derived Artin stacks parametrizing constructible sheaves on complex algebraic varieties and compact real analytic varieties. Furthermore, we show that every perversity function gives rise to an open substack of perverse…

Positivstellens\"atze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not…

We study the moduli stacks of real vector bundles of fixed rank and degree on a type I real algebraic curve and determine its mod 2 cohomology algebra in terms of characteristic classes.

The crystalline differential operators on a smooth variety X give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist X'. In some cases, this Azumaya algebra splits when restricted to finite covers of X'. In…