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Curves in projective space and RSK

This paper utilizes the RSK correspondence to provide a positive combinatorial interpretation of geometric Tevelev degrees for projective space, which enumerate pointed algebraic curves interpolating through maximal points, by expressing these invariants in terms of the combinatorics of words rather than Schubert calculus.

Original authors: Carl Lian, Saskia Solotko

Published 2026-02-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Carl Lian, Saskia Solotko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build a specific type of bridge (a mathematical curve) that connects a series of fixed landmarks (points in space). You have a strict budget (the degree of the curve) and a specific number of landmarks you must hit.

This paper is about counting exactly how many different ways you can build this bridge to hit all the required landmarks.

Here is the breakdown of the paper's story, translated from "Mathematician" to "Human":

1. The Problem: The "Point-Hitting" Puzzle

In the world of geometry, there's a famous puzzle: If you have a wiggly line (a curve) and you want it to pass through a specific set of dots in space, how many different lines can you draw?

The Setup: You have a curve with a certain amount of "wiggliness" (called genus). You have a 3D (or higher-dimensional) space. You pick a bunch of dots.
The Goal: You want to know the exact number of ways to draw a line of a specific size that hits every single dot.
The Catch: If you pick too many dots, it's impossible. If you pick just the right maximum number, there are a finite number of solutions. The authors call this number the Tevelev degree.

For a long time, mathematicians could calculate this number using very heavy, abstract algebra (like using a sledgehammer to crack a nut). They knew the answer, but they didn't have a simple, intuitive way to see why that was the answer.

2. The Old Way: The "Black Box" Calculator

Previous mathematicians solved this using something called Schubert Calculus. Think of this as a magical black box. You put the numbers (how many dots, how wiggly the curve is) into the box, and it spits out a number.

The Problem: The box works, but it doesn't tell you what those solutions actually look like. It's like knowing the answer to a riddle is "42," but not knowing the story behind it.

3. The New Way: The "Word Game" (RSK)

The authors, Carl Lian and Saskia Solotko, decided to open the black box. They used a famous mathematical tool called the RSK Correspondence (Robinson-Schensted-Knuth).

The Analogy:
Imagine you have a deck of cards. The RSK algorithm is a specific way of sorting those cards into two neat piles (tableaux).

The Magic: This sorting process creates a perfect link between shapes (the piles of cards) and words (sequences of letters).
The Breakthrough: The authors realized that every valid "bridge" (curve) corresponds to a specific word made of letters.

Instead of counting bridges, they are now counting words.

4. The Rules of the Word Game

The paper proves that the number of valid bridges is exactly equal to the number of words of a certain length that follow three simple rules:

The "Downhill" Rule: The word must contain enough "downhill" sections (decreasing sequences). If the word is too "flat" or "uphill," it doesn't represent a valid bridge.
The "Too Long" Rule: You can't have a section of the word that goes "uphill" (non-decreasing) for too long. If it gets too long, the bridge would be too stretched out to fit the budget.
The "Neighbor" Rule: You can't have a specific pattern where two neighboring letters (like 3 and 4) repeat too many times in a row. This prevents the bridge from getting stuck in a loop.

Why is this cool?
Before, the answer was a complex formula involving abstract shapes. Now, the answer is: "Count all the words of length X that don't break these three simple rules." It turns a high-level geometry problem into a combinatorial word puzzle.

5. The "Special Case" (When things get easy)

The paper also notes that if your bridge is very long (high degree), the rules become very simple.

The Analogy: If you have a very long rope, you can tie it to the dots in almost any order. The restrictions disappear.
The Result: In this easy case, the answer is just $(r+1)^g$ . It's like saying, "If you have 3 colors of paint and 5 spots to fill, you have $3^5$ ways to paint it." The authors show that their complex word rules naturally simplify to this easy answer when the bridge is long enough.

Summary

What did they do?
They took a difficult geometry problem about counting curves in space and translated it into a fun word game.

How did they do it?
They used a mathematical "translator" (the RSK correspondence) to turn geometric shapes into sequences of letters.

Why does it matter?
It gives us a positive interpretation. Instead of just getting a number from a black box, we can now visualize the solutions as specific patterns of letters. It connects the abstract world of shapes to the concrete world of words, making the math feel less like magic and more like a logical puzzle.

In a nutshell:

"We found that the number of ways to draw a wiggly line through a bunch of dots is exactly the same as the number of secret codes (words) you can write that follow three simple rules."

1. Problem Statement

The paper addresses the problem of computing geometric Tevelev degrees, denoted as $\text{Tev}^{\mathbb{P}^r}_{g,n,d}$ . These degrees enumerate the number of degree $d$ maps $f: C \to \mathbb{P}^r$ from a general smooth projective curve $C$ of genus $g$ to the $r$ -dimensional complex projective space, such that the map satisfies $n$ incidence conditions $f(p_i) = x_i$ for fixed general points $p_i \in C$ and $x_i \in \mathbb{P}^r$ .

The number of points $n$ is chosen to be the "maximal possible" for which the set of such maps is finite and transverse, given by the formula:
$n = \frac{r+1}{r}d - g + 1$
Prior to this work, these invariants were known to be expressible via Schubert calculus (intersection numbers on the Grassmannian $\text{Gr}(r+1, d+1)$ ), but a direct combinatorial interpretation of these counts was lacking, particularly outside the "large degree" regime ( $d \ge rg + r$ ).

2. Methodology

The authors employ a multi-step methodology bridging algebraic geometry and combinatorics:

L-tableaux Enumeration: They begin with the algebro-geometric characterization of $\text{Tev}^{\mathbb{P}^r}_{g,n,d}$ established in previous work (Lia24a), which expresses the count as the number of specific L-tableaux. An L-tableau consists of a grid filled with "blue" integers (forming a semi-standard Young tableau, SSYT) and "red" integers (forming a rotated SSYT), subject to specific constraints regarding "strips" (sequences of entries $i, i+1$ ).
Transformation to Tableau Pairs: Extending the work of Gillespie–Reimer-Berg (GRB23), the authors define a bijection between these constrained L-tableaux and pairs of tableaux $(P, Q)$ $(P, Q)$ of the same shape $\lambda$ $λ$ with $|\lambda| = g$ $∣ λ ∣ = g$ .
- $P$ is a semi-standard Young tableau (SSYT) with entries in $\{1, \dots, r+1\}$ .
- $Q$ is a standard Young tableau (SYT).
- The transformation involves "adjusting" the width of the blue tableau $B$ (adding or removing columns of height $r+1$ ) to handle cases where $d < g+r$ (the Brill-Noether special range).
RSK Correspondence: The authors apply the Robinson-Schensted-Knuth (RSK) correspondence to map the set of valid pairs $(P, Q)$ to $(r+1)$ -ary words of length $g$ .
Combinatorial Characterization: By analyzing the properties of the tableaux under the RSK map, they translate the geometric constraints on the L-tableaux (specifically the "1-strip-less" condition and width constraints) into conditions on the resulting words using:
- Greene's Theorem: Relating the shape of the tableaux to the lengths of disjoint increasing/decreasing subsequences.
- Knuth Equivalence: Analyzing how specific subsequence patterns (like $(i, i+1)$ -strips) behave under Knuth transformations.

3. Key Contributions

The primary contribution is a positive combinatorial proof of the geometric Tevelev degrees for all valid parameters $(g, r, d)$ , not just the large degree case.

Theorem 1.1 (Main Result): The authors prove that $\text{Tev}^{\mathbb{P}^r}_{g,n,d}$ is exactly equal to the number of $(r+1)$ -ary words of length $g$ satisfying three specific conditions:
1. Disjoint Decreasing Subsequences: The word contains at least $g + r - d$ disjoint decreasing subsequences of length $r+1$ .
2. Non-decreasing Bound: The word has no non-decreasing subsequence of length greater than $d/r$ .
3. Restricted $(i, i+1)$ Subsequences: The word has no $(i, i+1)$ -subsequence (a non-decreasing subsequence consisting only of $i$ and $i+1$ ) of length $d/r$ .
Unification of Regimes: The paper unifies the "Brill-Noether general" case ( $d \ge g+r$ ) and the "Brill-Noether special" case ( $d < g+r$ ). In the general case, the constraints become vacuous, recovering the known result $\text{Tev}^{\mathbb{P}^r}_{g,n,d} = (r+1)^g$ . In the special case, the constraints provide a precise reduction from the total $(r+1)^g$ words.
Extension of GRB23: The work generalizes the RSK-based bijection of Gillespie–Reimer-Berg, which was previously limited to the case $d \ge g+r$ , to the full range of geometric Tevelev degrees.

4. Results and Special Cases

Large Degree Limit: When $d \ge rg + r$ , the conditions in Theorem 1.1 are vacuous, and the count is simply $(r+1)^g$ . This matches the virtual enumeration in Gromov-Witten theory.
Sharp Bound: The paper demonstrates that the bound $d \ge rg + r$ is sharp. If $d < rg + r$, the count is strictly less than $(r+1)^g$ due to the non-vacuous constraints.
Case $d = rg$: The authors explicitly calculate the count for $d=rg$, showing it equals $(r+1)^g - n$ , where $n$ is the number of "disallowed" words (constant sequences or specific $(i, i+1)$ sequences). Geometrically, these disallowed words correspond to degenerate stable maps that contract the curve to a point.
Case $r=1$ (Plane Curves): For $r=1$ , the conditions simplify significantly. The problem reduces to counting binary words (or lattice paths) that do not cross a specific diagonal boundary. In the specific case $d = g/2 + 1$ , the result recovers the Catalan numbers, linking the Tevelev degree to the classical 19th-century calculation by Castelnuovo.

5. Significance

Combinatorial Interpretation: The paper provides the first purely combinatorial interpretation of geometric Tevelev degrees for arbitrary genus and degree, moving beyond intersection theory on Grassmannians.
Positive Formula: It offers a "positive" formula (counting specific objects) rather than an alternating sum or virtual intersection number, which is often difficult to interpret geometrically.
Bridge Between Fields: It strengthens the connection between enumerative geometry (Gromov-Witten theory, Tevelev degrees) and algebraic combinatorics (RSK, Young tableaux, Greene's theorem).
Geometric Insight: The combinatorial constraints (conditions i-iii) offer new insight into the geometry of the moduli space of maps, specifically characterizing the "degenerate" contributions that occur when the degree is low relative to the genus.

In summary, Lian and Solotko successfully translate a complex algebro-geometric counting problem into a tractable combinatorial problem involving words and subsequences, providing a complete and explicit formula for the geometric Tevelev degrees of projective space.