Research
I work in algebraic geometry, especially on questions connecting positivity with the equations and geometry of projective varieties, on the stability of vector bundles associated to embeddings, and on Hilbert schemes of points.
I study how positivity of an embedding line bundle governs projective normality, defining equations, and higher syzygies. A recurring goal is to understand which geometric features of a variety are visible in its homogeneous coordinate ring.
My recent work concerns the singularities and syzygies of higher secant varieties. This includes normality and Du Bois singularities, Cohen-Macaulay and rational singularities, symbolic powers, and analogues of gonality-type phenomena for embedded surfaces.
I am interested in linear series on curves and in the birational and projective geometry of their symmetric products, including divisor varieties and canonical embeddings of symmetric squares.
I am interested in the geometry of Hilbert schemes of points, including their components and singularities, and in deformation and smoothability questions for finite schemes.
I am interested in stability questions for syzygy and normal bundles, particularly on curves, as well as in tautological and Steiner bundles, Torelli questions, and related moduli problems.
Much of my present research asks how far familiar positivity-and-syzygy phenomena for curves extend to higher-dimensional varieties. Secant varieties provide a particularly rich testing ground: their geometry combines the positivity of the original embedding with subtle singularities and homological behavior.
I am also interested more broadly in Hilbert schemes of points, the geometry of curves and surfaces, the behavior and stability of vector bundles associated to embeddings, and problems in which algebraic information can be recovered from projective or moduli-theoretic constructions.