Geometric Tevelev Degrees

• Geometric Tevelev degrees are fixed-domain invariants that count morphisms from a fixed smooth pointed curve to a target variety under exact incidence conditions.
• They connect theories such as Hurwitz, Schubert calculus, and quantum cohomology, providing explicit formulas and recursions in various geometric settings.
• The framework distinguishes between geometric and virtual counts, with discrepancies arising from boundary phenomena and non-transversal map contributions.

Searching arXiv for papers on geometric Tevelev degrees and related tropical/logarithmic variants. Geometric Tevelev degrees are fixed-domain enumerative invariants attached to maps from a general pointed smooth curve to a target variety under the maximal number of point or incidence constraints compatible with a finite count. In the basic formulation, one studies the forgetful/evaluation morphism

$$\tau:\overline M_{g,n}(X,\beta)\to \overline M_{g,n}\times X^n$$

for a nonsingular projective target $X$ of dimension $r$, genus $g$, $n$ markings, and curve class $\beta$, under the expected-dimension condition

$\int_\beta c_1(T_X)=r(n+g-1).$

When every irreducible component dominating $\overline M_{g,n}\times X^n$ is generically smooth of expected dimension, the geometric Tevelev degree is the degree of $\tau$, equivalently the number of maps from a general $n$-pointed genus-$X$0 curve through $X$1 general target points (Lian et al., 2021, Lian, 2023).

1. Fixed-domain definition and conceptual scope

The defining feature of a geometric Tevelev degree is that the source curve has fixed complex structure. One fixes a general pointed smooth curve $X$2, chooses general target conditions, and counts morphisms $X$3 in class $X$4 satisfying those conditions. The count is expected to be finite precisely when

$X$5

which is the equality between the virtual dimension of $X$6 and the dimension of $X$7 (Lian et al., 2021, Buch et al., 2021).

This notion is distinct from the virtual Tevelev degree. The virtual invariant is defined by pushing forward the virtual fundamental class,

$X$8

whereas the geometric degree requires actual finiteness and transversality on a general fiber (Lian et al., 2021, Buch et al., 2021). A persistent theme in the literature is that virtual Tevelev degrees often admit very simple formulas, but geometric and virtual counts need not agree; discrepancies come from boundary stable maps, contracted components, rational tails, or other non-enumerative contributions (Lian, 2023, Lian et al., 2021).

Geometric Tevelev degrees count maps, not embedded image curves modulo reparametrization. If two distinct morphisms from the fixed source have the same image, they are counted separately. This map-counting perspective is fundamental in the fixed-domain interpolation problem and aligns Tevelev degrees with Brill–Noether theory, linear series, Hurwitz theory, and the geometry of evaluation maps on moduli spaces (Lian et al., 2021, Lian, 2023).

2. Hurwitz-theoretic origins and the $X$9 case

The original Tevelev degree arises from Hurwitz moduli. For degree-$r$0 simply branched covers $r$1 of genus $r$2 with $r$3 marked unramified points, the admissible-cover compactification $r$4 carries a natural morphism

$r$5

In Tevelev’s original situation,

$r$6

this map is finite, and the degree

$r$7

counts degree-$r$8 maps from a general $r$9-pointed genus-$g$0 curve to a general $g$1-pointed rational curve, up to the ordering of the $g$2 simple ramification points. The resulting formula is

$g$3

(Cela et al., 2021).

The Hurwitz-space interpretation extends to the two-parameter family

$g$4

For these parameters the source and target of the corresponding Hurwitz forgetful map still have equal dimension, and the generalized degrees $g$5 count covers in which $g$6 distinguished marked points lie in one common fiber. The central recursion is

$g$7

obtained by excess intersection on boundary strata of admissible-cover spaces. In the classical sector $g$8 this specializes to $g$9; in other regimes it interpolates with projective geometry of lines and with Castelnuovo’s count of pencils (Cela et al., 2021).

For $n$0, the theory admits a further geometric generalization allowing arbitrary ramification profiles over marked target points and equal-image constraints among marked source points. If $n$1 are ramification profiles over $n$2 marked target points, with

$n$3

the generalized degree

$n$4

counts admissible covers with prescribed grouped ramification, where

$n$5

A basic structural simplification is that the degree depends on each $n$6 only through its total size $n$7. The theory admits both a genus recursion and explicit closed formulas, and it can also be expressed by Schubert-calculus formulas on $n$8 via limit linear series (Cela et al., 2021).

3. Projective-space geometric Tevelev degrees

For $n$9, the fixed-domain point-interpolation problem is especially rigid. If $\beta$0 is a general pointed smooth curve of genus $\beta$1, and $\beta$2 are general points, the geometric Tevelev degree

$\beta$3

is the number of degree-$\beta$4 maps $\beta$5 with $\beta$6, under the balance

$\beta$7

(Lian, 2023, Lian et al., 27 Aug 2025).

The projective-space case is now known in all degrees allowed by this dimension constraint. The main formula is an explicit Grassmannian intersection: $\beta$8 obtained by degenerating the fixed source curve and replacing naïve map spaces by spaces of complete collineations. This resolves excess intersections coming from linear dependence and base-point phenomena and yields a complete geometric answer for point conditions in $\beta$9, as well as a complete answer in $\int_\beta c_1(T_X)=r(n+g-1).$0 with arbitrary point and line conditions (Lian, 2023).

The projective-space theory sharply distinguishes geometric from virtual counts. For $\int_\beta c_1(T_X)=r(n+g-1).$1, the virtual answer is

$\int_\beta c_1(T_X)=r(n+g-1).$2

when the dimension constraint holds, but geometric and virtual degrees differ outside the large-degree regime. In $\int_\beta c_1(T_X)=r(n+g-1).$3, the first non-large-degree case $\int_\beta c_1(T_X)=r(n+g-1).$4 has

$\int_\beta c_1(T_X)=r(n+g-1).$5

and for $\int_\beta c_1(T_X)=r(n+g-1).$6 one gets

$\int_\beta c_1(T_X)=r(n+g-1).$7

(Lian, 2023).

A later combinatorial reformulation uses the RSK correspondence. Under

$\int_\beta c_1(T_X)=r(n+g-1).$8

$\int_\beta c_1(T_X)=r(n+g-1).$9 equals the number of $\overline M_{g,n}\times X^n$0-ary words of length $\overline M_{g,n}\times X^n$1 satisfying three subsequence conditions: there exist at least $\overline M_{g,n}\times X^n$2 disjoint decreasing subsequences of length $\overline M_{g,n}\times X^n$3; there is no nondecreasing subsequence of length $\overline M_{g,n}\times X^n$4; and there is no $\overline M_{g,n}\times X^n$5-subsequence of length $\overline M_{g,n}\times X^n$6. This converts the Schubert formula into a manifestly positive word count and explains why the large-degree regime collapses to

$\overline M_{g,n}\times X^n$7

(Lian et al., 27 Aug 2025).

4. Virtual theory, enumerativity, and logarithmic variants

Virtual Tevelev degrees admit a uniform quantum-cohomological expression. For a nonsingular projective target $\overline M_{g,n}\times X^n$8, point class $\overline M_{g,n}\times X^n$9, and quantum Euler class

$\tau$0

the virtual invariant satisfies

$\tau$1

This reduction to small quantum cohomology yields exact formulas for projective spaces, quadrics, cominuscule homogeneous spaces, and low-degree complete intersections (Buch et al., 2021).

Enumerativity is a separate issue. A central asymptotic result proves that for fixed genus and sufficiently positive curve class, virtual Tevelev degrees are enumerative for all homogeneous varieties and for hypersurfaces of sufficiently low degree compared to the dimension. The mechanism is twofold: smooth-domain maps in a general fiber become unobstructed and transverse, while boundary strata with rational tails fail to dominate the incidence target. In genus $\tau$2, stronger non-asymptotic statements hold in the same classes (Lian et al., 2021).

For hypersurfaces, there is also a direct geometric computation. If $\tau$3 is a smooth hypersurface of degree $\tau$4, with

$\tau$5

then

$\tau$6

for all $\tau$7 if $\tau$8, and for sufficiently large $\tau$9 if $n$0. This reproves, by projective geometry, the asymptotic geometric count previously obtained by combining virtual formulas with enumerativity results (Lian, 2022).

Logarithmic variants enlarge the theory to toric targets with tangency conditions. For a toric variety $n$1, discrete data $n$2, and the logarithmic moduli space $n$3, one studies

$n$4

In genus $n$5, under the fixed-domain dimension condition

$n$6

the virtual logarithmic Tevelev degree equals the actual number of logarithmic stable maps from a general fixed source curve, provided boundary and toric-fixed-point degeneracies are excluded. For Hirzebruch surfaces $n$7, the genus-$n$8 tropical and logarithmic counts coincide and admit explicit closed formulas in the contact partitions $n$9 (Cela et al., 2023).

For Fano complete intersections of dimension $X$00, the quantum Euler class has been computed explicitly, leading to formulas for all virtual Tevelev degrees in classes $X$01 satisfying

$X$02

That work does not prove general geometric enumerativity, but it gives the full virtual answer and an algorithm reducing the remaining computation to genus-$X$03 two-point hyperplane invariants (Cela, 2022).

5. Tropical Tevelev degrees and correspondence theorems

The tropical counterpart is defined by replacing admissible covers with tropical admissible covers and replacing Deligne–Mumford spaces with tropical moduli spaces. In the classical Hurwitz case

$X$04

one has a finite tropical forgetful map

$X$05

and the tropical Tevelev degree is

$X$06

Because the simple branch ends are already unmarked on the tropical side, there is no $X$07 denominator. The main tropical theorem is

$X$08

proved by a correspondence theorem via Berkovich skeletons together with a direct combinatorial count of tropical admissible covers (Cavalieri et al., 2024).

The tropical computation is highly explicit. One chooses a special point in

$X$09

whose source curve has a chain of $X$10 loops and whose target is a caterpillar-like tree. Each loop contributes one of exactly two allowed local fragments, $X$11 or $X$12, altering the active-path degree by $X$13. The genus part is therefore encoded by lattice paths that never cross below $X$14, and the marked-tree part is reconstructed by cut-and-join operations along the active path. A binomial recursion counts the genus possibilities, and every resulting tropical cover has multiplicity $X$15, so the degree is literally the number of covers (Cavalieri et al., 2024).

This tropical framework has recently been generalized by introducing the parameter $X$16, so that

$X$17

For positive $X$18, the tropical degree remains unchanged: $X$19 For negative $X$20, one obtains an explicit defect formula: $X$21 The same paper also defines generalized tropical Tevelev degrees with ramification profiles $X$22, giving closed formulas that tropicalize the generalized algebraic degrees studied for $X$23 (Dawson, 29 Jan 2026).

A recurring conceptual point is that tropical Tevelev degrees are not merely analogous to the algebraic ones. In the classical case $X$24 is proved by tropicalization of moduli spaces and comparison of local analytic and tropical degrees, and the later generalized work is explicitly framed as tropicalizing the algebraic theories of Cela–Pandharipande–Schmitt and Cela–Lian (Cavalieri et al., 2024, Dawson, 29 Jan 2026).

6. Vanishing phenomena, exceptional targets, and adjacent degree theories

Geometric Tevelev degrees are not uniformly positive. A sharp vanishing result holds for Hirzebruch surfaces

$X$25

If the expected-dimension condition

$X$26

holds and $X$27 (or $X$28 and $X$29), then no irreducible component of $X$30 dominates $X$31, except in the single case

$X$32

Hence, outside that case,

$X$33

The proof uses semistability of $X$34 for general counted maps as a necessary condition for interpolation (Cela et al., 26 Mar 2025).

The exceptional case reduces exactly to $X$35. For

$X$36

one has

$X$37

Geometrically, a map $X$38 in class $X$39 is equivalent to a degree-$X$40 pencil $X$41 together with a section of $X$42, and the marked-point conditions determine that section uniquely. Thus the only nonzero Hirzebruch-surface counts for $X$43 are inherited from the already-computed geometric Tevelev degrees of $X$44 (Cela et al., 26 Mar 2025).

A broader but related multidegree theory is provided by Kapranov degrees on $X$45. These are top intersections of divisor classes

$X$46

equivalently multidegrees of the product of forgetful and Kapranov maps. They include the Castravet–Tevelev case as a special situation, and their positivity is characterized by the Cerberus condition

$X$47

In the specialization $X$48 for all $X$49, the Kapranov degree is either $X$50 or $X$51, recovering the Castravet–Tevelev phenomenon inside a larger multidegree formalism (Brakensiek et al., 2023).

Taken together, these developments show that geometric Tevelev degrees form a moduli-theoretic family of fixed-domain interpolation invariants with several distinct faces: Hurwitz-theoretic for $X$52, Schubert-theoretic for projective space, quantum-cohomological on the virtual side, logarithmic for toric targets with tangency, and tropical through correspondence theorems. The subject is marked by two complementary facts: in some regimes the answers are unexpectedly rigid, such as $X$53 or $X$54; in others, boundary contributions, semistability obstructions, or ramification constraints force genuine deviations from the virtual or large-degree picture (Lian, 2023, Lian et al., 2021).