arXiv subjects
Publications and source records attributed to Alessio Cela.
We tropically enumerate planar well-spaced elliptic curves with a fixed $j$-invariant. Combined with the recent genus 1 correspondence theorem of Cela--Koyama, this yields an alternative proof of Pandharipande's algebraic enumeration.
We compute the K-theory of weighted blowups of smooth stacks satisfying the resolution property along smooth centers. As an application, we determine the K-theory of the stack of stable genus 1 curves with 2 marked points. Furthermore, we express the Lambda polynomial of the tangent complex of the blowup morphism in terms of the blowup data. Along the way, we generalize the Anderson-Payne construction of equivariant operational K-theory from torus actions to actions of smooth affine algebraic groups.
We show that the genuinely enumerative count of algebraic elliptic curves in any toric variety agrees with the count of the corresponding well-spaced tropical curves, weighted by explicit combinatorial multiplicities. This provides a complete genus-$1$ generalization of the celebrated Nishinou--Siebert correspondence theorem in genus $0$. The proof is algebro-geometric and relies on logarithmic deformation theory together with an explicit enumeration of logarithmic maps with fixed tropicalization.
In arbitrary characteristic, we determine the maximum number of general points through which a rational curve of degree $d$ in $\mathbb{P}^r$ passes, subject to an additional secancy condition along a linear space. We consider the cases both where the points on the curve are unprescribed and prescribed, which amount to the determination of the normal and restricted tangent bundles of a general rational curve in $\mathsf{Bl}_{\mathbb{P}^s}\mathbb{P}^r$, respectively. In the appendix, we enumerate the interpolating curves in the case of prescribed points on the curve.
This paper is the third and final part of a series devoted to the description of the integral Chow rings of the moduli stacks of hyperelliptic Prym pairs. For a fixed genus $g$, there are two natural stacks, $\mathcal{RH}_g$ and $\widetilde{\mathcal{RH}}_g$, parametrizing hyperelliptic Prym pairs, with the former being the $\mu_2$-rigidification of the latter. Both decompose as the disjoint union of $\lfloor (g+1)/2 \rfloor$ components, denoted $\mathcal{RH}_g^n$ and $\widetilde{\mathcal{RH}}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper we present quotient stack descriptions of the components $\mathcal{RH}_g^n$ for even $g$ and compute their integral Chow rings, thereby completing the computation for all irreducible components of $\mathcal{RH}_g$. In addition, we give quotient stack presentations for all irreducible components of $\widetilde{\mathcal{RH}}_g$ and determine when the rigidification map $\widetilde{\mathcal{RH}}_g^n \to \mathcal{RH}_g^n$ is a root gerbe. We then use this to compute the Chow rings of $\widetilde{\mathcal{RH}}_g^n$ for all $g$ and $n$, with the sole exception of the case where $g$ is odd and $n=(g+1)/2$. Finally, in the appendix, we discuss $G$-gerbes induced by an homomorphism of abelian groups $H \to G$ and an $H$-gerbe.
This paper is the second in a series devoted to describing the integral Chow ring of the moduli stacks $\mathcal{RH}_g$ of hyperelliptic Prym pairs. For fixed genus $g$, the stack $\mathcal{RH}_g$ is the disjoint union of $\lfloor (g+1)/2 \rfloor$ components $\mathcal{RH}_g^n$ for $n = 1, \ldots, \lfloor (g+1)/2 \rfloor$. In this paper, we give presentations and compute the integral Chow rings of the components $\mathcal{RH}_g^{(g+1)/2}$ for odd $g$. As an application, we also obtain presentations and Chow rings for all irreducible components of the moduli stack of hyperelliptic Spin curves of odd genus. An intermediate result of independent interest is the computation of the integral Chow ring of the moduli stack of unordered pairs of divisors of the same even degree in $\mathbb{P}^1$.
We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.
We prove a symplectic version of a conjecture of Lian and Pandharipande: in sufficiently high degree, the fixed-domain Gromov-Witten invariants of positive symplectic manifolds are signed counts of pseudo-holomorphic curves. The original conjecture in the complex algebraic setting was recently disproved by Beheshti et al. However, we show that the statement holds when the complex structure is replaced by a generic almost complex structure. The proof relies on showing that the fixed-domain Gromov-Witten pseudocycle can be constructed without the use of inhomogeneous or domain-dependent perturbations, which answers positively a question posed by Ruan and Tian.
When $a\ge2$, we show that a general pointed curve never interpolates through the expected number of points in the Hirzebruch surface $\mathcal{H}_a$, with one exception. In the exceptional case, the number of such interpolating maps is determined by the geometric Tevelev degrees of $\mathbb{P}^1$, which have been previously computed.
The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth, projective surface. Namely, a non-constant map deforms in a family of expected dimension as long as its image has anti-canonical degree at least 4. In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms certain toric contractions. We raise the question of whether a Brill-Noether theorem could hold for toric varieties of higher dimension.
This paper is the first in a series dedicated to computing the integral Chow rings of the moduli stacks of Prym pairs. In this work, we compute the Chow ring for Prym pairs arising from a single pair of Weierstrass points and from at most $(g-1)/2 $ pairs when the genus $g$ of the curve is odd.
In this paper we compute the integral Chow ring of the moduli stack $\mathcal{R}_2$ of Prym pairs of genus 2 with integral coefficients.
We compute the rational Chow ring of the moduli stack of planar nodal curves of fixed degree and express it in terms of tautological classes. Along the way, we extend Vial's results on Chow groups of Brauer-Severi varieties to $G$-equivariant settings.
For a non-singular projective toric variety $X$, the virtual logarithmic Tevelev degrees are defined as the virtual degree of the morphism from the moduli stack of logarithmic stable maps $\overline{\mathcal{M}}_{\mathsfΓ}(X)$ to the product $\overline{\mathcal{M}}_{g,n} \times X^n$. In this paper, after proving the genus $0$ correspondence theorem in this setting, we use tropical methods to provide closed formulas for the case in which $X$ is a Hirzebruch surface. In order to do so, we explicitly list all the tropical curves contributing to the count.
We study the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and virtual (Gromov-Witten) counts are found to agree asymptotically in the Fano (and some $(-K)$-nef) examples, but not in general. For toric blow-ups, geometric counts are expressed in terms of integrals on products of Jacobians and symmetric products of the domain curves, and evaluated explicitly in genus 0 and in the case of $\text{Bl}_q(\mathbb{P}^r)$. Virtual counts for $\text{Bl}_q(\mathbb{P}^r)$ are also computed via the quantum cohomology ring.
Let $(C,p_1,\ldots,p_n)$ be a general curve. We consider the problem of enumerating covers of the projective line by $C$ subject to incidence conditions at the marked points. These counts have been obtained by the first named author with Pandharipande and Schmitt via intersection theory on Hurwitz spaces and by the second named author with Farkas via limit linear series. In this paper, we build on these two approaches to generalize these counts to the situation where the covers are constrained to have arbitrary ramification profiles: that is, additional ramification conditions are imposed at the marked points, and some collections of marked points are constrained to have equal image.
We compute the quantum Euler class of Fano complete intersections X in a projective space. In particular, we prove a recent conjecture of A. Buch and R. Pandharipande. Finally we apply our result to obtain formulas for the virtual Tevelev degrees of X. An algorithm computing all genus 0 two-point Hyperplane Gromov Witten invariants of X is illustrated along the way.
We interpret the degrees which arise in Tevelev's study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalization). We find exact solutions which specialize to Tevelev's formula in his cases and connect to the projective geometry of lines and Castelnuovo's classical count of linear series in other cases. For almost all values, the calculation of the two parameter generalization of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.