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Adrian Zahariuc

Publications and source records attributed to Adrian Zahariuc.

13 recordsLinked to original sources
arXiv2025

Small resolutions of moduli spaces of scaled curves

We construct small resolutions of the moduli space $\overline{Q}_n$ of stable scaled $n$-marked lines of Ziltener and Ma'u--Woodward and of the moduli space $\overline{P}_n$ of stable $n$-marked ${\mathbb G}_a$-rational trees introduced in earlier work. The resolution of $\overline{P}_n$ is the augmented wonderful variety corresponding to the graphic matroid of the complete graph. The resolution of $\overline{Q}_n$ is a further blowup, also a wonderful model of an arrangement in ${\mathbb P}^{n-1}$.

math.AG↗
arXiv2025

Multiscale differentials and wonderful models

We study the relationships between several varieties parametrizing marked curves with differentials in the literature. More precisely, we prove that the space $\mathcal{B}_n$ of multiscale differentials of genus 0 with $n+1$ marked points of orders $(0,\ldots,0,-2)$ is a wonderful variety. This shows that the Chow ring of $\mathcal{B}_n$ is generated by the classes of a collection of smooth boundary divisors with normal crossings subject to simple and explicit linear and quadratic relations. Furthermore, we realize $\mathcal{B}_n$ as a subvariety of the space $\mathcal{A}_n$ of multiscale lines and prove that $\mathcal{B}_n$ can be realized as the normalized Chow quotient of $\mathcal{A}_n$ by a natural $\mathbb{C}^*$-action.

math.AG↗
arXiv2024

Marked nodal curves with vector fields

We discuss two operations on nodal curves with (logarithmic) vector fields, which resemble the `stabilization' construction in Knudsen's proof that $\bar{\mathcal M}_{g,n+1}$ is the universal curve over $\bar{\mathcal M}_{g,n}$. We prove that both operations work in families (commute with base change). We construct inverse operations under suitable assumptions, which allow us to prove a technical result quite similar to Knudsen's, in the case of curves with vector fields. As an application, we prove that the Losev--Manin compactification of the space of configurations of $n$ points on ${\mathbb P}^1 \backslash \{0,\infty\}$ modulo scaling degenerates isotrivially to a compactification of the space of configurations of $n$ points on ${\mathbb A}^1$ modulo translation, and the natural group actions fit together globally.

math.AG↗
arXiv2024

Interpolation of fat points on K3 and abelian surfaces

We prove that any number of general fat points of any multiplicities impose the expected number of conditions on a linear system on a smooth projective surface, in several cases including primitive linear systems on very general K3 and abelian surfaces, `Du Val' linear systems on blowups of ${\mathbb P}^2$ at $9$ very general points, and certain linear systems on some ruled surfaces over elliptic curves. This is done by answering a question of the author about the case of only one fat point on a certain ruled surface, which follows from a circle of results due to Treibich--Verdier, Segal--Wilson, and others.

math.AG↗
arXiv2024

Configurations of points on a line up to scaling or translation

We prove that the Losev--Manin compactification of the space of configurations of $n$ points on ${\mathbb P}^1 \backslash \{0,\infty\}$ modulo scaling degenerates (isotrivially) to a compactification of the space of configurations of $n$ points on ${\mathbb A}^1$ modulo translation. The latter resembles the compactification constructed by Ziltener and Mau--Woodward, but allows the marked points to coincide, making it a ${\mathbb G}_a^{n-1}$-variety, which mirrors the fact that the Losev--Manin space is toric. The degeneration is compatible with the actions of ${\mathbb G}_m^{n-1}$ and ${\mathbb G}_a^{n-1}$ in the sense that these actions fit together globally in the total space of the degeneration.

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arXiv2024

Elliptic surfaces and linear systems with fat points

We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a certain polarization, we observe that the special case of only one fat point implies the general case of arbitrarily many fat points, as well as results concerning other surfaces. We conjecture that this special case holds in characteristic 0, but prove that it fails in any positive characteristic.

math.AG↗
arXiv2019

A Riemann-Hurwitz-Plucker formula

We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety. Our definition of ramification is relative to an algebraic family of divisors on the target variety, and our formula is obtained using the theory of refined top Chern classes. In assigning multiplicities to ramification points, we frequently have to consider excess degeneracy loci, but we are able to show nonetheless that the multiplicities are always nonnegative, and are positive under very mild hypotheses.

math.AG↗
arXiv2018

Reducible Calabi-Yau threefolds with countably many rational curves

We give a class of examples of reducible (d-semistable) threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such stable maps occur in infinitely many degrees.

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arXiv2017

Deformation of Quintic Threefolds to the Chordal Variety

We consider a family of quintic threefolds specializing to a certain reducible threefold. We describe the space of genus zero stable morphisms to the central fiber (as defined by J. Li). As an elementary application of an extension of the analysis, we prove the existence of rigid stable maps of arbitrary genus and sufficiently high degree to very general quintics.

math.AG↗
arXiv2017

Rational Curves on Del Pezzo Manifolds

We exploit an elementary specialization technique to study some properties of rational curves on index $n-1$ Fano $n$-folds. We prove a simple formula for counting rational curves passing through a suitable number of points in the case $n=3$. The arguments have immediate deformation theoretic consequences which translate to properties of the normal bundles of such curves.

math.AG↗
arXiv2017

The irreducibility of the generalized Severi varieties

We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -- are irreducible.

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