Papers and preprints

Given a Bridgeland stability condition on a 2-Calabi–Yau category, we define a simplicial complex that encodes the Harder–Narasimhan filtrations of spherical objects. For 2-Calabi–Yau categories of type \(A\), we relate this complex to the complex of pointed pseudo-triangulations on configurations of points on the plane. Using this connection, we prove that the complex undergoes piecewise-linear wall-crossings as we vary the stability condition, and is piecewise-linearly homeomorphic to a sphere.

Additionally, we prove that for a generic stability condition on a 2-Calabi–Yau category, a spherical object is determined by the ordered list of its Harder–Narasimhan factors.

Let \(X\) be an analytic K3 surface with \(\operatorname{Pic} X = 0\). We describe the closure of the Bridgeland stability manifold of \(X\) obtained using the masses of semi-rigid objects.

We describe Baily-Borel, toroidal, and geometric — using the KSBA stable pairs — compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order \(3\) and \(4\) for which the fixed locus of the automorphism contains a curve of genus \(\ge2\). For order \(3\), we treat all the maximal-dimensional such families. We show that the toroidal and the KSBA compactifications in these cases admit simple descriptions in terms of certain \(ADE\) root lattices.

We study the syzygies of the canonical embedding of a ribbon \(\widetilde{C}\) on a curve \(C\) of genus \(g \geq 1\). We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus \(p_a\) on a general curve of genus \(g\) with \(p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}\). Among non-general ribbons, the case of split ribbons is particularly interesting.

Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for \(C\) and it implies Green’s conjecture for all double covers \(C'\) of \(C\) with \(g(C') \geq \textrm{max}\{3g+2, 6g-4\}\). We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to \(C\), which may be of independent interest.

This paper culminates in the count of the number of 3-Veronese surfaces passing through 13 general points. This follows the case of 2-Veronese surfaces discovered by Coble in the 1920’s. One important element of the calculation is a direct construction of a space of “complete triangles.” Our construction is different from the classical ordered constructions of Schubert, Collino and Fulton, as it occurs directly on the Hilbert scheme of length 3 subschemes of the plane. We transport the enumerative problem into a 26-dimensional Grassmannian bundle over our space of complete triangles, where we perform Atiyah-Bott localization. Several important questions arise, which we collect at the end of the paper.

We compute equivariant fundamental classes of orbits in GL(2)-representations. As applications, we find degrees of the orbit closures corresponding to elliptic fibrations and self-maps of the projective line.

We compute the equivariant fundamental class of the orbit closure of a linear series on the projective line. We also describe the boundary of the orbit closure and how the orbits specialise in one parameter families.

Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and that a general cubic surface arises 42120 times as a hyperplane section of a general cubic 3-fold.

Consider a 2-Calabi–Yau triangulated category with a Bridgeland stability condition. We devise an effective procedure to reduce the phase spread of an object by applying spherical twists. Using this, we give new proofs of the following theorems for 2-Calabi–Yau categories associated to ADE quivers: (1) all spherical objects lie in a single orbit of the braid group, and (2) the space of Bridgeland stability conditions is connected.

Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory, and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.

We describe a compactification of the moduli space of pairs (S,C) where S is a smooth quadric surface in 3-space and C is a smooth divisor on S of class (3,3). We show that the compactified moduli space is a smooth Deligne-Mumford stack with 4 boundary components. We relate our compactification with compactifications of the moduli space of genus 4 curves. In particular, we show that our space compactifies the blow-up of the hyperelliptic locus. We also relate our compactification to a compactification of the Hurwitz space of triple coverings of the projective line by genus 4 curves.

We propose compactifications of the moduli space of Bridgeland stability conditions of a triangulated category. Our construction arises from a viewing a stability condition as a metric on the underlying category and is inspired by the Thurston compactification of the Teichm\“uller space of hyperbolic metrics on a surface. The key ingredient in the construction are maps from the stability manifold to an infinite projective space. We prove that, under suitable hypotheses, these maps are injective and their image has a compact closure. We identify a family of points in the boundary that are categorical analogous to the intersection functionals in Teichm\”uller theory.

We study in detail the geometry of the resulting compactification for the 2-Calabi–Yau categories of quivers, and fully work out the cases of the \(A_2\) and \(\widehat{A_1}\) quivers. To do so, we carefully examine the dynamics of Harder–Narasimhan multiplicities under auto-equivalences of the category. We introduce a finite automaton to study this dynamics and employ it in our analysis of the \(A_{2}\) and \(\widehat{A_1}\) categories.

We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties. Going further, we study the map that associates to a linear projection its ramification divisor. We show that this map is dominant for most (but not all!) varieties of minimal degree, using (linked) limit linear series of higher rank. We find the degree of this map in some cases, extending the classical appearance of Catalan numbers in the geometry of rational normal curves, and give a geometric explanation of its fibers in terms of torsion points of naturally occurring elliptic curves in the case of the Veronese surface and the quartic rational surface scroll.

The classical statement of Cayley-Salmon that there are 27 lines on every smooth cubic surface in \(\mathbf{P}^3\) fails to hold under tropicalization: a tropical cubic surface in \(T\mathbf{P}^3\) often contains infinitely many tropical lines. Under mild genericity assumptions, we show that when embedded using the Eckardt triangles in the anticanonical system, tropical cubic del Pezzo surfaces contain exactly 27 tropical lines. In the non-generic case, which we identify explicitly, we find up to 27 extra lines, no multiple of which lifts to a curve on the cubic surface. We realize the moduli space of stable anticanonical tropical cubics as a four-dimensional fan in \(\mathbf{R}^40\) with an action of the Weyl group \(W(E_6)\). In the absence of Eckardt points, we show the combinatorial types of these tropical surfaces are determined by the boundary arrangement of 27 metric trees corresponding to the tropicalization of the classical 27 lines on the smooth algebraic cubic surfaces. Tropical convexity and the combinatorics of the root system \(E_6\) play a central role in our analysis.

We construct a well-behaved compactification of the space of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in \(\overline{M}_6\) of the locus of smooth plane quintic curves.

We describe a sequence of effective divisors on the Hurwitz space H(d,g) for d dividing g-1 and compute their cycle classes on a partial compactification. These divisors arise from vector bundles of syzygies canonically associated to a branched cover. We find that the cycle classes are all proportional to each other.

We prove the analogue for ribbons of Green\’s canonical syzygy conjecture, formulated by Bayer and Eisenbud. Our proof uses the results of Voisin and Hirschowitz-Ramanan on Green\’s conjecture for general smooth curves.

We introduce the problem of GIT stability for syzygy points of canonical curves with a view toward a GIT construction of the canonical model of the moduli space of stable curves. As the first step in this direction, we prove semi-stability of the first syzygy point for a general canonical curve of odd genus.

We prove that the rational Picard group of the simple Hurwitz space H_d,g is trivial for d up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of nodal curves on Hirzebruch surfaces.

We use Groebner basis techniques to study the balanced canonical ribbon in each odd genus g ≥ 5. We obtain equations and syzygies of the ribbon, give a Groebner interpretation of part of Alper, Fedorchuk, and Smyth\’s proof of finite Hilbert stability for canonical curves, and discuss the obstacles in using ribbons to give a new proof of Generic Green\’s Conjecture (Voisin\’s Theorem).

We construct several modular compactifications of the Hurwitz space \(H^d_{g/h}\) of genus \(g\) curves expressed as $d$-sheeted, simply branched covers of genus \(h\) curves. These compactifications are obtained by allowing the branch points of the covers to collide to a variable extent. They are very well-behaved if \(d = 2, 3\), or if relatively few collisions are allowed. We recover as special cases the spaces of twisted admissible covers of Abramovich, Corti and Vistoli and the spaces of hyperelliptic curves of Fedorchuk.

We algebraically compute the class of the Hodge eigenbundles in the cyclic covering construction using Grothendieck-Riemann-Roch for stacks.

We establish sharp bounds for the slopes of curves in \(\overline{M}_g\) that sweep the locus of trigonal curves, proving Stankova-Frenkel\’s conjectured bound of 7+6/g for even g and obtaining the bound 7+20/(3g+1) for odd g. For even g, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd g, we describe the analogous extremal effective divisor and give a similar explicit expression.

We construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent. Beginning with the standard admissible cover compactification, the sequence first proceeds through contractions of the boundary divisors and then through flips of the so-called Maroni strata, culminating in a Fano model for even genera and a Fano fibration for odd genera. While the sequence of divisorial contractions arises from a more general construction, the sequence of flips uses the particular geometry of triple covers. We explicitly describe the Mori chamber decomposition given by this sequence of flips.

We construct several modular compactifications of the Hurwitz space \(H^d_{g/h}\) of genus \(g\) curves expressed as $d$-sheeted, simply branched covers of genus \(h\) curves. They are obtained by allowing the branch points of the cover to collide to a variable extent, generalizing the spaces of twisted admissible covers of \citet*{acv:03}. The resulting spaces are very well-behaved if \(d\) is small or if relatively few collisions are allowed. In particular, for \(d = 2\) and \(3\), they are always well-behaved. For \(d = 2\), we recover the spaces of hyperelliptic curves of \citet{fedorchuk10:_modul_hyp}. For \(d = 3\), we obtain new birational models of the space of triple covers.

We describe in detail the birational geometry of the spaces of triple covers of \(\P^1\) with a marked fiber. In this case, we obtain a sequence of birational models that begins with the space of marked (twisted) admissible covers and proceeds through the following transformations:

1. sequential contractions of the boundary divisors,
2. contraction of the hyperelliptic divisor,
3. sequential flips of the higher Maroni loci,
4. contraction of the Maroni divisor (for even \(g\)).

The sequence culminates in a Fano variety in the case of even \(g\), which we describe explicitly, and a variety fibered over \(\P^1\) with Fano fibers in the case of odd \(g\).

. We study the process of normalization of affine algebraic varieties and an algorithm to carry it out. We illustrate the application of normalization to the problem of resolution of singularities by proving that normal varieties are regular in codimension 1.