arXiv subjects

Carl Lian

Publications and source records attributed to Carl Lian.

At least 19 recordsLinked to original sources

arXiv2026

Interpolation for rational curves with secants

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.

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arXiv2025

Curves in projective space and RSK

The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending ideas of Gillespie--Reimer-Berg, we use the RSK correspondence to give a positive interpretation of these counts in terms of the combinatorics of words.

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arXiv2025

Delta-matroids and toric degenerations in OG(n,2n+1)

We construct an explicit, embedded degeneration of the general torus orbit closure in the maximal orthogonal Grassmannian OG(n,2n+1) into a union of Richardson varieties. In particular, we deduce a formula for the cohomology class of the torus orbit closure, as a sum of products of Schubert classes. The moment map images of the degenerate pieces are the base polytopes of their underlying delta-matroids, and give a polyhedral decomposition of the unit hypercube, which had previously been studied by Chen-Sanchez-Veliz-Ying.

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arXiv2025

Enumerating log rational curves on some toric varieties

The genus 0, fixed-domain log Gromov-Witten invariants of a smooth, projective toric variety X enumerate maps from a general pointed rational curve to a smooth, projective toric variety passing through the maximal number of general points and with prescribed multiplicities along the toric boundary. We determine these invariants completely for the projective bundle X=P_{P^r}(O^s+O(-a)), proving a conjecture of Cela--Iribar L\'opez. A different conjecture when X is the blow-up of P^r at r points is disproven. Whereas the conjectures were predicted using tropical methods, we give direct intersection-theoretic calculations on moduli spaces of "naive log quasimaps."

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arXiv2025

Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

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.

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arXiv2025

Curves on Hirzebruch Surfaces and Semistability

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.

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arXiv2025

A Brill-Noether Theorem for (toric) surfaces

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.

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arXiv2024

Modularity of $d$-elliptic loci with level structure

We consider the generating series of special cycles on $\mathcal{A}_1(N)\times \mathcal{A}_g(N)$, with full level $N$ structure, valued in the cohomology of degree $2g$. The modularity theorem of Kudla-Millson for locally symmetric spaces implies that these series are modular. When $N=1$, the images of these loci in $\mathcal{A}_g$ are the $d$-elliptic Noether-Lefschetz loci, which are conjectured to be modular. In the appendix, it is shown that the resulting modular forms are nonzero for $g=2$ when $N\geq 11$ and $N\neq 12$.

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arXiv2024

$d$-elliptic loci and the Torelli map

We show that two natural cycle classes on the moduli space of compact type stable maps to a varying elliptic curve agree. The first is the virtual fundamental class from Gromov-Witten theory, and the second is the Torelli pullback of the special cycle on A_g of principally polarized abelian varieties admitting an elliptic isogeny factor.

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arXiv2023

On the asymptotic enumerativity property for Fano manifolds

We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree $d \leq (n+3)/3$ in $\mathbb{P}^n$.

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arXiv2023

The HHMP decomposition of the permutohedron and degenerations of torus orbits in flag varieties

Let $Z\subset Fl(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson-Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada-Horiguchi-Masuda-Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson-Tymoczko formula. We construct an explicit toric degeneration inside $Fl(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson-Tymoczko formula.

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arXiv2023

Degenerations of complete collineations and geometric Tevelev degrees of $\mathbb{P}^r$

We consider the problem of enumerating maps $f$ of degree $d$ from a fixed general curve $C$ of genus $g$ to $\mathbb{P}^r$ satisfying incidence conditions of the form $f(p_i)\in X_i$, where $p_i\in C$ are general points and $X_i\subset\mathbb{P}^r$ are general linear spaces. We give a complete answer in the case where the $X_i$ are points, where the counts, the ``Tevelev degrees'' of $\mathbb{P}^r$, were previously known only when $r=1$, when $d$ is large compared to $r,g$, or virtually in Gromov-Witten theory. We also give a complete answer in the case $r=2$ with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations.

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arXiv2023

Torus orbit closures and 1-strip-less tableaux

We compare two formulas for the class of a generic torus orbit closure on the Grassmannian, due to Klyachko and Berget-Fink. The naturally emerging combinatorial objects are semi-standard fillings we call 1-strip-less tableaux.

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arXiv2023

Enumerativity of virtual Tevelev degrees

Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus $g$ maps of fixed complex structure in a given curve class $β$ through $n$ general points of a target variety $X$. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class $β$. A simple speculation is that for all Fano $X$, the virtual Tevelev degrees are enumerative for sufficiently large $β$. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension). As an application, we prove a new result on the existence of very free curves of low degree on hypersurfaces in positive characteristic.

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arXiv2023

Fixed-domain curve counts for blow-ups of projective space

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.

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arXiv2023

Asymptotic geometric Tevelev degrees of hypersurfaces

Let $(C,p_1,\ldots,p_n)$ be a fixed general pointed curve and let $(X,x_1,\ldots,x_n)$ be a smooth hypersurface of degree $e$ and dimension $r$ with $n$ general points. We consider the problem of enumerating maps $f:C\to X$ of degree $d$ (as measured in the ambient projective space) such that $f(p_i)=x_i$. When $e$ is small compared to $r$ and $d$ is large compared to $g$, $e$, and $r$, these numbers have been computed first by passing to a virtual count in Gromov-Witten theory obtained by Buch-Pandharipande, and then by showing (in work of the author with Pandharipande) that the virtual counts are enumerative via an analysis of boundary contributions in the moduli space of stable maps. In this note, we give a simpler computation via projective geometry.

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arXiv2022

Generalized Tevelev degrees of $\mathbb{P}^1$

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.

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arXiv2022

Linear series on general curves with prescribed incidence conditions

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We determine these numbers completely for linear series of arbitrary dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or $n=r+2$. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt.

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