[Paper Review] On the enumeration of rational plane curves with tangency conditions

Charles Cadman|ArXiv.org|Sep 28, 2005

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper develops a recursive formula to enumerate rational plane curves of degree $d$ that pass through specified points on a smooth cubic $E \subset \mathbb{P}^2$, have prescribed tangency conditions (order 1 and order 2) at specified and unspecified points on $E$, and pass through additional general points in $\mathbb{P}^2$. Using twisted stable maps to a stack $\mathbb{P}^2_{E,2}$ constructed via the square root construction along $E$, the author generalizes Kontsevich's recursion and extends Caporaso-Harris-type formulas to cubic curves, proving enumerativity of Gromov-Witten invariants in this setting.

ABSTRACT

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass through 3d-1-a-2b-c general points of P^2? The answer is given as a generalization of Kontsevich's recursion. We also investigate more general enumerative problems of this sort, and prove an analogue of a formula of Caporaso and Harris.

Motivation & Objective

• To solve the enumerative problem of counting rational degree $d$ curves in $\mathbb{P}^2$ with specified and unspecified tangency conditions to a smooth cubic $E$.
• To extend Kontsevich's recursion for rational curves through general points to include tangency constraints with $E$.
• To establish that Gromov-Witten invariants on the stack $\mathbb{P}^2_{E,2}$ are enumerative, capturing curves with prescribed contact orders.
• To generalize the Caporaso-Harris formula to the case of contact with a smooth plane cubic rather than a line or conic.
• To provide a recursive algorithm for computing the number of such curves, implementable via a Maple program.

Proposed method

• The paper uses twisted stable maps to the Deligne-Mumford stack $\mathbb{P}^2_{E,2}$, obtained by applying the square root construction along a smooth cubic $E \subset \mathbb{P}^2$.
• It defines contact types via the $\mu_2$-action on the stack, which encode the order of tangency of a curve with $E$ at marked points.
• The enumeration is reduced to computing intersection numbers on the moduli space of twisted stable maps to $\mathbb{P}^2_{E,2}$, leveraging the fact that maps with even-order contact lifts to the stack.
• A deformation theory argument is used to show that only components with expected dimension contribute to Gromov-Witten invariants, ensuring enumerativity.
• The recursion is derived by analyzing the behavior of maps under node separation and using induction on the number of nodes in the domain curve.
• The key technical tool is the proof that evaluation maps on the moduli stack are generically surjective, and that non-expected-dimensional components do not contribute to invariants.

Experimental results

Research questions

• RQ1 How many rational degree $d$ curves in $\mathbb{P}^2$ pass through $a$ general points on a smooth cubic $E$, have $b$ specified order-2 tangency conditions with $E$, $c$ unspecified order-2 tangency conditions, and $3d-1-a-2b-c$ general points in $\mathbb{P}^2$?
• RQ2 Can Kontsevich's recursion for rational curves through general points be generalized to include tangency conditions with a smooth plane cubic?
• RQ3 Is the Gromov-Witten invariant counting curves with specified contact orders to a cubic $E$ enumerative, i.e., equal to the actual number of such curves?
• RQ4 What is the structure of the moduli space of twisted stable maps to $\mathbb{P}^2_{E,2}$, and how does it relate to classical enumerative geometry?
• RQ5 Can a Caporaso-Harris-type recursion be established for contact with a smooth plane cubic, analogous to the case of lines and conics?

Key findings

• The number $N_d(a,b,c)$ of rational degree $d$ curves satisfying the specified tangency and point conditions is given by a generalized recursion that extends Kontsevich’s formula.
• The Gromov-Witten invariants of the stack $\mathbb{P}^2_{E,2}$ are enumerative, meaning they count actual curves satisfying the geometric constraints.
• The recursion is derived by analyzing the deformation theory of twisted stable maps and using induction on the number of nodes in the domain curve.
• Evaluation maps on the moduli stack of twisted stable maps are generically surjective, ensuring that the expected dimension condition is sufficient for enumerativity.
• The formula accounts for both specified and unspecified tangency points, with specified contacts requiring fixed points on $E$ and unspecified ones allowed to vary.
• A Maple program implementing the recursion is available from the author’s homepage, enabling explicit computation of $N_d(a,b,c)$ for small $d$.