Space curves and solitons of the KP hierarchy: I. The $l$-th generalized KdV hierarchy

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\ldots, lm+k\rangle$ where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.


Introduction
The purpose of this paper is to identify algebraic curves associated with particular class of soliton solutions of the KP hierarchy. It is known that solutions of the KP equation can be constructed from any algebraic curves [15]. A solution from a smooth curve is a quasi-periodic solution, and some soliton solutions can be constructed by a rational (singular) limit of the curve with only ordinary double points (see e.g. [19,28,3]). In particular, the cases corresponding to the KdV and nonlinear Schödinger equations are well-studied, in which the algebraic curves are given by the hyperelliptic curves (see e.g. [3,19]). Recently, there are several papers dealing with some non-hyperelliptic cases, e.g. so-called (n, s)-curves, where the authors construct the Klein σ-functions over these curves (see e.g. [5,6,12,13,14,17,20]). It seems, however, that almost no result has been reported for the cases with more general algebraic curves. Because of the difficulty in finding a canonical homological basis for the general algebraic curves, it may be quite complicated to compute explicitly a rational limit of these curves and the corresponding Riemann theta functions (see [20]). On the other hand, a large number of real regular soliton solutions of the KP hierarchy has been classified in terms of finite Grassmann manifolds Gr(N, M ) (see e.g. [11,10,9]). Recently, there are some progress on the study concerned with the connections between the algebraic-geometric solutions and these soliton solutions [1,2,21,22].
In this paper, we take the first step to identify explicit forms of algebraic curves associated with certain class of KP solitons. We consider complex (singular in general) soliton solutions in terms of the complex Grassmannians. Here what we mean by "soliton" is that the underlying M -dimensional space C M of the Grassmannian for the solution is spanned by only "exponential" functions. We also introduce the "generalized" soliton solutions which are mixed with rational functions in addition to the exponential functions (see [21]), which are obtained by multiple degenerations of the soliton parameters. In particular, choosing a special point of the Schubert cell of the Grassmannian, the τ -function corresponding to a generalized soliton has a Schur polynomial expansion whose leading polynomial is determined by the Young diagram corresponding to the Schubert cell. The main tool in this paper is the Sato universal Grassmannian which is the moduli space of the formal power series solutions of the KP hierarchy [26,27,21,22].
We briefly give a background of the Sato theory of the KP hierarchy and the algebro-geometric solutions associated with algebraic curves. Sato's theory of KP hierarchy associates a point of the universal Grassmannian manifold (UGM), a solution of the KP hierarchy. Each solution of the KP hierarchy is given by the so-called τ -function, which is a function of infinitely many variables (t n : n = 1, 2 . . .). It is known that an algebro-geometric solution of the KP hierarchy can be expressed by the Riemann theta function [15,3,18]. A method of algebro-geometric solutions was proposed in [15], and these solutions are constructed by the following way. Let C be a compact Riemann surface of genus g and L a holomorphic line bundle of degree g over it. Let p ∞ be a nonsingular point of C, z a local parameter around p ∞ . Let H 0 (C, L( * ∞)) be meromorphic sections of L with poles only at p ∞ . Identifying L with a degree g divisor D on C, this can be represented as a space of meromorphic functions U on a small neighborhood of p ∞ . Such U represents a point of UGM, and denoted by U = Span C {B} with the Sato frame B representing a basis of the space of meromorphic functions with poles at the specified divisors D + * ∞ on C. In the present paper, we only consider the set of meromorphic functions with a pole only at p ∞ , i.e. D = gp ∞ . (The general case will be discussed in a future communication.) Let W = H 0 (C, O( * ∞)) be the set of meromorphic functions on C with a pole only at p ∞ . We note that W is a commutative algebra graded by the order of poles at p ∞ and U is a module over W , where we have z g W = U ∈ UGM [28]. We also note that the order of poles of elements in W defines a numerical semigroup S.
It turns out that the algebraic curve associated with the generalized soliton solution discussed in this paper is a "singular space curve". The strategy to find the curve is as follows: (1) Given a τ -function associated with a generalized soliton solution, we identify the corresponding Sato frame B by embedding the solution into UGM as a point U = Span C {B}. (2) We define a basis A for a maximal graded commutative subalgebra R in C[z −1 ], over which U is an R-module. The basis A is filtered by the order of poles at p ∞ , and in the cases under consideration it has the form A = z −g f (z)B with some polynomial f (z) with f (0) = 1. (3) We identify a (singular) space curve C whose affine part is given by Spec(R). In the case of a smooth curveC, there is an identification, Span C {A} = H 0 (C, O( * ∞)) and z g H 0 (C, O( * ∞)) ∈ UGM. The numerical semigroup S represents the order of poles of the basis A, and the genus of S is the arithmetic genus of the curve C.
The present paper is organized as follows. In Section 2, we define a class of generalized soliton solutions of the KP hierarchy which are defined by further degenerations of the soliton parameters. These solitons are particular solutions of the l-th generalized KdV hierarchy for some integer l ≥ 2, which is also referred to as the l-reduction of the KP hierarchy defined by ∂τ /∂t nl = c n τ with some constants c n for all n = 1, 2, . . .. In particular, we consider the generalized soliton solutions from the center elements of the Schubert cell of the Grassmannian Gr(N, M ) (Theorem 2.2).
In Section 3, we briefly review the notion of Sato universal Grassmannian (UGM) and give an explicit embedding of the generalized soliton solutions into UGM. Each soliton solution is then expressed as a point of UGM (Proposition 3.1).
In Section 4, we study the numerical semigroups of type l, lm + 1, lm + 2, · · · , lm + k for m ≥ 1 and 1 ≤ k ≤ l − 1, and present the corresponding Young diagrams, which will be used to parametrize the generalized soliton solutions from the l-th generalized KdV hierarchy. We also give the explicit formulas of the Frobenius number and (arithmetic) genus of these numerical semigroups (Proposition 4.1).
In Section 5, we give the main theorem (Theorem 5.1), which states that algebraic curves for the generalized soliton solutions are singular space curve in C k+1 associated with the numerical semigroup of type l, lm + 1, · · · , lm + k . Theorem 5.1 also provides the Schur expansions of the corresponding τ -function. We then present several explicit examples to illustrate the theorem.
In Section 6, we give the proof of Theorem 5.1. This section also provides a summary of our main results.
In Section 7, we study a deformation of the singular space curve in C l for the soliton solution associated to the numerical semigroup S = l, lm+1, . . . , l(m+1)−1 (Proposition 7.1). It is then shown that these soliton solutions are obtained by coalescing two moduli parameters of the corresponding smooth curve. The singularity of the singular space curve is an ordinary l-tuple point singularity (see e.g. [7]), and in the case with l = 2, i.e. a plane curve, it is an ordinary "double point" singularity. As a byproduct, we show that the numerical semigroup of type l, lm + 1, . . . , l(m + 1) − 1 is a Weierstrass semigroup for m ≥ 2. We also give a brief discussion on deformation of singular curves for the generalized soliton solutions associated with the types l, lm + 1, . . . , lm + k with 1 ≤ k < l − 1.
Finally, in Section 8, we reconstruct the singular space curves for the soliton solution from the point of view of commutative differential operators as in the work of Burchnall and Chaundy [4] (Theorem 8.5).
In the appendix, we provide some brief background information about the Lax-Sato formulation of the KP hierarchy, the l-th generalized KdV hierarchy and the soliton solutions. useful discussion on the singularity of space curves. The present work is partially supported by NSF grant DMS-1714770.

The generalized soliton solutions and the Schur expansion of the τ -function
The KP equation is a nonlinear partial differential equation in the form, The KP equation admits an infinite number of symmetries (commuting flows), and these flows are parametrized by the "time" variables t n for n ∈ N. The set of all commuting flows is called the KP hierarchy, whose first member is the KP equation.
The KP soliton solutions are classified in terms of finite dimensional Grassmannians. Here we briefly summarize these previous results for the background of the present paper (also see Appendix A).
2.1. The τ -functions for the generalized soliton solutions. The τ -function of the KP hierarchy with the time variables t = (t 1 , t 2 , . . .) is introduced by The soliton solutions are constructed as follows: Let {f i (t) : 1 ≤ i ≤ N } be a set of linearly independent functions f i (t) of infinite variables t = {t n : n ∈ N} satisfying the following system of linear equations, The τ -function is then given by the Wronskian with respect to t 1 -variable, (See (A.9) in Appendix A, and also [9] for the details.) As a fundamental set of the solutions of (2.3), we take the exponential functions, where κ j 's are arbitrary constants. For the soliton solutions, we consider f i (t) as a linear combination of the exponential solutions, where A := (a i,j ) is an N × M constant matrix of full rank, rank(A) = N , Then the τ -function (2.4) is expressed by where A T is the transpose of the matrix A, and E(t) is given by Note here that the set of exponential functions {E 1 (t), . . . , E M (t)} gives a basis of M -dimensional space of the null space of the operator M i=1 (∂ 1 − κ i ), and we call it a "basis" of the KP soliton. Then the set of functions {f 1 (t), . . . , f N (t)} represents an N -dimensional subspace of M -dimensional space spanned by the exponential functions. This leads naturally to the structure of a finite Grassmannian Gr(N, M ), the set of N -dimensional subspaces in C M . Then the N × M matrix A of full rank can be identified as a point of Gr(N, M ). The Gr(N, M ) has a decomposition called the Schubert decomposition given by (2.8) Gr where λ is a Young diagram contained in an N × (M − N ) rectangular diagram denoted by (M − N ) N , and X λ is the Schubert cell which consists of the N × M matrices whose pivots are given by In terms of the pivot indices, the Young diagram λ = (λ 1 , . . . , λ N ) is expressed as That is, the pivot indices of the matrices in X λ appear at the vertical paths in a lattice path starting from the top right conner and ending at the bottom left corner in the Young diagram λ as shown in the figure above.
We also have the following formula of the τ -function using the Binet-Cauchy lemma (see e.g. [9]),  2.2. The Schur function expansion of the τ -function. Expanding the exponential function E j (t) with respect to the powers of κ j , we have where p n (t) is the elementary Schur polynomials. That is, E j (t) is the generating function of the elementary Schur polynomials, which are given by Note that the elementary Schur polynomials satisfy ∂ k p n (t) = ∂ k 1 p n (t) = p n−k (t) and p m (t) = 0 if m < 0. Using the elementary Schur polynomials, the τ -function can be expressed as follows: First note that where A T is the transpose of the N × M matrix A. Expanding the exponential functions, we have Note that each function f i (t) in (2.5) has the following expansion, The τ -function (2.6) is then expressed by 1 p 1 p 2 · · · · · · · · · · · · 0 1 p 1 p 2 · · · · · · · · · . . . . . .
In this paper, we include the generalized bases which contain derivatives of the exponential functions with respect to the parameters κ j [21]. For example, we consider the following derivative, Note thatÊ (0) j = E j . Then one can take the following basis, where q j 's are nonnegative integers. The τ -function in this case can be expressed in the form (2.11) by replacing the Vandermonde matrix K in (2.10) with which we call the generalized Vandermonde matrix. Then the functionsf i (t) for the τ -function, denoted byτ = Wr(f 1 , . . . ,f N ), has the expansion, a i,j q j + n n κ n j p n (t).
Remark 2.1. The soliton solutions generated by the τ -function with the basis (2.12) contain rational functions of the variables t = (t 1 , t 2 , . . .), hence they are singular in general.
Now we consider the expansion of the τ -function, which we call the Schur expansion of the τ -function: Using the Binet-Cauchy lemma, one can expand the τ -function (2.4) in terms of the Schur polynomials, where S λ (t) is the Schur function associated with the Young diagram λ = (λ 1 , λ 2 , . . . , λ N ) defined by The constant cμ in (2.16) are the N × N minors of the ∞ × N matrix KA T (orKA T ). In particular, the first constant c λ , which is normalized to be one, is the minimal minor of KA T (orKA T ) in the lexicographical order of the row indices {l 0 , l 1 , . . . , l N −1 } for 0 ≤ l 0 < l 1 < · · · < l N with the relation, With this relation, the Schur polynomial associated with the Young diagram λ is given by Notice that in (2.16) if A is totally nonnegative, then λ = ∅ and τ (0) = 1 (note that a real KP soliton is regular, if and only if A is totally nonnegative [11]). Here, we are interested in the singular solutions where τ (0) = 0. In particular, we look for the expansion (2.16) with a particular matrix A so that the leading Schur polynomial has the largest possible weight. Recall that the weight of S λ (t) is defined by the number of boxes in λ, denoted by |λ| = N i=1 λ i . Note that the weight of t n is n.
We then have the following theorem: Theorem 2.2. Fixed a Schubert cell X λ ⊂ Gr(N, M ), there exists a unique element A ∈ X λ , so that the leading term of the Schur expansion of the τ -function (2.11) generated by A and K in (2.10) (orK in (2.14)) is S λ (t). That is, the pivot indices, say Proof. Assume A is in an row echelon form (REF), i.e.
Remark 2.3. One should note in general that for an A ∈ X λ , the τ -function has a Schur expansion with the leading polynomial S µ (t), whose Young diagram satisfies µ ⊂ λ. In the case of λ = µ in Theorem 2.2, the unique matrix A (more precisely, the reduced row echelon form of A ∈ X λ ) is sometimes referred to as the "center" of the cell X λ . Note that the center A also depends on K (orK). where r i 's are some constants. We consider the following set of f i 's: Let {n 1 , . . . , n m } be a partition of N with 1 ≤ n s < l, i.e. N s = n 1 + · · · + n s . Take Here we have r is = κ l s and ω l = exp(2πi/l), the l-th root of unity. Then the N × ml matrix A consists of m diagonal blocks in the form, Note that the matrix A corresponds to the base function, (E 1 (t, κ 1 ), . . . , E l (t, κ 1 ), E 1 (t, κ 2 ), . . . , E l (t, κ 2 ), . . . , . . . , . . . , E 1 (t, κ m ), . . . , E l (t, κ m )) , which we simply write as With the reduction (2.19), we have wheret is the set of time variables without t nl for n = 1, 2, . . .. Then the τ -function (2.4) is expressed by κ nl s t nl Wr(g 1 , . . . , g N )(t).
Since the solution of the KP hierarchy is given by (2.2), the exponential factor of the τ -function does not give any contribution to the solution u. Then the τ -function, denoted by τ l (t), for the l-reduced KP hierarchy can be expressed by The generalized soliton solutions for the l-th generalized KdV hierarchy. One should first note that the reduction condition (2.19) is not compatible with the derivatives (2.12), that is, the derivativê E (q) does not satisfy the condition (2.19) for q > 0. In order to construct the generalized soliton solutions for the reduced equation, we consider a limit κ i → κ j for fixed κ j . For the sake of simplicity, we take κ 1 = κ 2 + ∆κ. Then the elements in the first l terms in the set (2.22), that is, the basis is given by The τ -function is then given by that is,Ã is obtained by adding the first block of l columns to the second block of l columns and removing the first block of l columns. And τ 1 (t) is a τ -function generated by the same matrix A and the new basis, Then we have the following proposition.
Proposition 2.4. If the rank ofÃ in (2.24) is less than N , then the τ -function generated by the basis given in (2.25) and the matrix A in (2.21) gives a solution of the l-th generalized KdV hierarchy.
Proof. The rank condition, rank(Ã) < N , implies τ 0 (t) = 0. Then from (2.23), we have the limit, This means that τ 1 (t) generated by the base (2.25) and the matrix A in (2.21) is a solution of the l-reduced KP hierarchy.
With the rank condition in Proposition 2.4, the following degeneration of the κ-parameters leads to the change of the basis (see (5.1) below), It is then easy to see that a further confluence of the κ-parameters leads to the generalized bases (2.13), and appropriate rank conditions on the matrix A generate a τ -function for the KP hierarchy.
Remark 2.5. If the matrix A satisfies the rank conditions given in Proposition 2.4, one can use the derivative ∂ ∂κ (κE(t, κ)) instead of ∂ ∂κ E(t, κ). Note that the formula (2.12) is useful for the Schur expansion of the τ -function as we will see in the later sections. We denote this generalized base by (2.26) [ . We will give several explicit examples in Section 5.

The Sato universal Grassmannians
Here we give a brief description of the Sato universal Grassmannian (see e.g. [21,27] for further details). Let V = C((z)) be the vector space of formal Laurent series, which has a decomposition, Let π be the projection map, Then the Sato universal Grassmannian (UGM) is defined by Each element of UGM is expressed by Here the base function φ −i (z) has the following form: for some integer N ≫ 0, Here the columns are labeled by −N 0 from right to left as shown in (· · · , −2, −1, 0), and the rows are indexed in downward by Z, as in the transpose of (· · · , −2, −1, 0, 1, 2, · · · ) The submatrix with * 's in the bottom right section corresponds to the coefficients (b j,−i ) with 0 ≤ i ≤ N − 1 and j ≥ −N + 1.
Note that the top row with * 's is at j = −N + 1, and the left most column with * 's is at i = N − 1, that is, the top left conner is at (−N + 1, −N + 1), marked by * . LetB be a column echelon form of the matrixB, and writeB ≡B, i.e. they are column-equivalent.
Note that m −i represents the pivot index of the column The set m is expressed by the so-called Maya diagram as follows: For the case with m i = −i for all i ∈ N 0 (this is called the vacuum state), theB is a lower triangular matrix whose diagonals are all 1's, The corresponding Maya diagram is then given by the horizontal boxes filled with the black balls as shown below, where the index of the box for the right most black ball is "0", and the index of each box increases toward right. The vacuum state corresponds to the element with N = 0. Now taking rightmost N black balls and relocating them so that the boxes filled by the black balls have the index set This can be considered as a Schubert decomposition of UGM. For example, the index set m = (7, 4, 2, 0, −2, −5, −6, · · ·) gives the case N = 5, which corresponds to the following Maya diagram, Here the index of the right most black ball is "7". One should note that the number of filled boxes with the positive indices is the same as that of empty boxes with nonpositive indices. This implies the condition for UGM, i.e. dim ( It is also known that the Maya diagram is alternatively expressed by the Young diagram. For the above example, we have Using the Young diagrams, we can have the Schubert decomposition of UGM similar to (2.8), where Y is the set of all Young diagrams, and each element in the Schubert cellX λ associated with 3.1. UGM for the generalized soliton solutions. Let us identify each soliton solution as a point of UGM. More precisely, we consider an embedding associated with the matrix K in (2.10) (orK in (2.14) for the generalized soliton), Here the submatrix marked with * 's is given by the ∞ × N matrix KA T (orKA T ), and the top left conner as marked by * has the coordinates (−N + 1, −N + 1) ∈ Z × (−N 0 ). We then denoteB by KA T . Since the basis B is uniquely determined by the Sato frameB, we sometimes refer to B as the Sato frame. We then have the following proposition.
where c i is a normalization constant, and [z] represents Proof. First we have the following formulas (see e.g. [9]), where∂ is defined by That is, the operator exp( 1 n z n ∂ n ) replaces p m (t) with z m , i.e.
From the formulas of the τ -function (2.11) and the functionf i (t) in (2.15), note that the function φ −i (z) is obtained by replacing p n (t) in (2.15) with z n . Then placef 1 (t) at the (−N + 1)-th column place in the Sato frame, so that the expansion off 1 (t) starts with the factor z −N +1 . This implies the formula in the proposition. The constant c i normalizes the leading term in the expansion of φ −i (z) to be one.
We note that the exponential functionsÊ In conclusion, the embedding σK for the generalized soliton solution gives σK (A) = Span C {B} ∈ UGM with a basis B ⊂ C((z)), where c i 's are chosen so that each φ −i (z) is a monic function in C((z)). We also remark thatf i ([z]) is a rational function with some power of (1 − (κ j z) l ) −1 . Then multiplying B by that is, each element ofB is a Laurent polynomial. We further multiplyB by z −g for some positive integer g, so that we have a subset of polynomials in C[z −1 ] including 1, i.e.
which is a central object in this paper. Then the main theorem in this paper (Theorem 5.1) shows that the subset A for a certain class of soliton solutions spans a subalgebra in C[z −1 ].
Remark 3.2. The subsetB also gives a point of UGM, which is different from the point given by B.
However, the solution u of the KP hierarchy remains the same for both B andB. This can be seen as follows.
we have the following formula for the functionĝ That is, we useĝ i (t) for the τ -function instead off i (t). Then the τ -function from the functions which implies that the solution u = 2∂ 2 1 ln τ is invariant under this multiplication (also see [29]). 3.2. The soliton solutions of the l-th generalized KdV hierarchy. We here consider the following simple but useful example of the soliton solution associated with Gr(1, l). From (2.19) and (2.3), we have r = κ l , and the κ-parameters are given by We take the matrix A = (1, ω l , ω 2 l , . . . , ω l−1 l ) ∈ Gr(1, l) and the ∞ × l Vandermonde matrix K in (2.10), The following notations are useful: Let Ω k l be a vector given by (3.10) which satisfy the orthogonality relations, Let us take the matrix A as Then we have which leads to the expansion form of the τ -function, Note that the corresponding soliton solution is singular, and the leading Schur function, p l−1 (t), has the Young diagram with l − 1 boxes in horizontal.
Multiplying B by z −l+1 (1 − (κz) l ), we have a basis of a subspace of the set of functions having pole only at z = 0, We then note that the set A represents a commutative ring of polynomials defined on a certain (singular) algebraic curve. In order to show this, we introduce the following variables, (3.13) x = z −l , and Then A gives a set of meromorphic functions having pole only at ∞ in these variables, Note here that we have the following relations among the variables (x, y 1 , . . . , y l−1 ), where we define y 0 = F (x) = x − κ l and y l = G(x) = xF (x). These p i,j can be expressed as the 2 × 2 minors of the 2 × l matrix, (3.15) y 1 y 2 · · · y l−1 G(x) F (x) y 1 · · · y l−2 y l−1 This expression will be very useful when we consider the deformation of the singular curves associated with these relations in Section 7.
It is then important to notice that the vector space generated by the setÃ in (3.14) can be expressed by a commutative ring R of the variables (x, y 1 , . . . , y l−1 ), i.e.
where the prime ideal is given by the minors, That is, the variables (x, y 1 , . . . , y l−1 ) are the coordinates for a singular curve C, whose affine part is given by Spec(R).

The numerical semigroups for the reductions
Let us start with a brief introduction to the numerical semigroups (see, e.g. [25] for the details). It is known that every numerical semigroup is generated by a finite set {x 1 , x 2 , . . . , x n } ⊂ N, i.e.
It is also well known that S is a numerical semigroup if and only if gcd{x 1 , . . . , x n } = 1. Every numerical semigroup has a unique finite set which generates S. We also have the following notions for the numerical semigroup; F (S) · · · · · · · · · g m Here 0, s i , s j ∈ S and g m , F (S) ∈ N 0 \ S.
Proposition 4.1. The Frobenius number F (S) and the genus g(S) of the numerical semigroup S = l, lm + 1, lm + 2, . . . , lm + k are given by where n l,k = l − 1 k .
Proof. Let us first consider the case of m = 1. Since l is the smallest generator of S and l + k is the largest one, there exists the largest number denoted by n l,k such that l(n + 1) − (l + k)n > 1 for n ≤ n l,k . This largest number n l,k is given by n l,k = l−1 k . This means that any number a ≥ ln n,k is in S, and we have F (S) = ln l,k − 1.
Let [a, b] denote the set of numbers {a, a + 1, . . . , b} for a < b ∈ N 0 . Since the gaps are the integers in the set, , the number of gaps are counted by For the case with m > 1, we have m copies of the above situation and obtain the formulas asserted in the proposition. This is easy to see from the Young diagram given in the previous section. Also from the proof of Proposition 4.1, we have that for m = 1, the genus g(S) has a partition, The following lemma is easy to prove. Then the Young diagram λ(S) for S = l, l + 1, l + 2, . . . , l + k (m = 1) has the following shape: Each rectangular box (shaded region in the figure below) at the southeast boundary has the size g i (S)×n i (S), i.e. g i (S) boxes in horizontal, and n i (S) in vertical, as shown below.  That is, the Young diagram λ(S) = (λ 1 , λ 2 , . . . , λ N ) is given by n i for k = 1, . . . , n l,k and N = mN n l,k . Likewise, in the vertical direction, we have We take the base functions (2.26) with the following set of exponentials, and E(κ i ) = (E 1 (t, κ i ), . . . , E l (t, κ i )) with E p (t, κ i ) = exp( With the expansion κ j E(t, κ) = ∞ n=0 κ j+n p n (t), we havê is an ∞ × l matrix whose n-th row (counting from n = 0) is the l-dimensional vector given by with Ω n l = (1, ω n l , ω 2n l , . . . , ω n(l−1) l ). Then the τ -function (2.11) of the generalized soliton solution of the l-reduction is given by whereK is given by an ∞ × lmq matrix, 2 , . . . ,K (j) m ). Then we have the following main theorem in this paper.
Theorem 5.1. Let (l, m, k) be a triplet of numbers satisfying l ≥ 2, m ≥ 1 and 1 ≤ k ≤ l − 1. Consider a generalized soliton solution based on the exponential functions (5.1) with q = n l,k = ⌈ l−1 k ⌉ and the A-matrix given by where I m is the m × m identity matrix, n i = k(i − 1) + 1, and [Ω l ] n is an n × l matrix defined by Then we have the followings: Here λ a,b is λ a,b = (m − b + 1)g a + mg a+1 + · · · + mg q , with g a = l − n a = l − 1 − k(a − 1). (b) The image of the embedding σK (A) ∈ UGM withK in (5.2) generates a numerical semigroup of type S = l, lm + 1, lm + 2, . . . , lm + k with the Young diagram λ(S) = λ. (c) The generalized soliton solution generated by the τ -function (5.3) is then associated with an irreducible component of the following singular space curve, (d) The τ -function of (5.3) has the following Schur expansion, wheret = {t n : n ≡ 0 (mod l)} and cμ(κ) is a homogeneous symmetric polynomial of {κ l i : i = 1, . . . , m}, Here the Young diagram λ is given by λ(S) of the numerical semigroup S, and each coefficient cμ is uniquely determined by µ.
We give the proof of Theorem 5.1 in Section 6. In the next subsections, we give some examples to illustrate the theorem.
The soliton solution we consider here has the following κ-parameters and the m × 2m matrix A ∈ Gr(m, 2m), where Ω 1 2 = (1, −1) and 0 2 = (0, 0). The matrix K is an ∞ × 2m matrix, Through the embedding σ K in (3.3), the image σ K (A) gives the basis {φ −i : i ∈ N 0 } in (3.1), We then normalize the set by multiplying z −g m i=1 (1 − (κ i z) 2 ) with g = m. Then we have a basis of a subspace in C Note that the first m terms can be expressed simply by the following set, the set of meromorphic functions having pole only at ∞, 1, z −2 , . . . , z −2m+2 . Then this basis can be expressed in a simple form, We then introduce the following variables, which gives the relation defining a singular hyperelliptic curve, Then the set A in the variables (x, y), denoted byÃ, gives a coordinate ring of the curve, All the elements in R are graded and are given in the table, 0 1 2 3 4 · · · · · · 2m 2m + 1 2m + 2 2m + 3 · · · 1 * x * x 2 · · · · · · x m y x m+1 xy · · · which gives the numerical semigroup of type 2, 2m + 1 . Thus, the soliton solutions of the 2-reduction (KdV reduction) correspond to the "double" point degeneration of the smooth hyperelliptc curve, The degenerated curve C is then given by the double point limit, This is a well-known result showing that each KdV soliton solution is obtained by a double point limit of the theta function associated with the hyperelliptic curve [19,28]. Also note that we have We remark here that the type of the numerical semigroup remains the same as that of the singular curve.
We have the following theorem of the Schur expansion of the τ -function: Theorem 5.2. The τ -function associated with the numerical (hyperelliptic) semigroup S = 2, 2m + 1 has the expansion where λ = (m, m − 1, . . . , 1) and each µ is obtained by adding a diagram consisting of several tiles to λ. The coefficients cμ(κ) are the Schur polynomials associated to the Young diagramsμ determined by the structure of the added diagram µ in terms of tiles, that is,μ is obtained by adding to the previous diagram in the same manner as in the diagram µ (see the example below).
Proof. From the form of KA T in (5.6) and using the Binet-Cauchy formula, the proof is obvious. See the example 5.3 and Lemma 5.4 below.
For the general m, the coefficients cμ(κ) withμ = (μ 1 ,μ 2 , . . . ,μ m ) is given by the Schur polynomial, where h α (κ) is the complete homogeneous symmetric polynomials of {κ 2 1 , κ 2 2 , . . . , κ 2 m }. To show this, we need the following lemma (equivalent to the Giambelli's formula): Lemma 5.4. The reduced row echelon form of the m×∞ Vandermonde matrix V = (x j−1 i ) for 1 ≤ i ≤ m and 1 ≤ j is given by Proof. First recall that the Schur polynomial c µ (x) of (x 1 , . . . , x m ) can be also defined by the Jacobi's bialternant formula, where λ = (λ 1 , . . . , λ m ). The (r, s)-element of the row reduced echelon form V −1 0 V for 1 ≤ r ≤ m and s > m is given by which is expressed by the Schur polynomial c (s−m−1|m−r) (x). That is, we have , which is the desired formula.
The Young diagram is then given by for the case with m = 3. The soliton solution here has the following κ-parameters and the m × 3m matrix A, where Ω k 3 := (1, ω k 3 , ω 2k 3 ) with ω 3 = exp 2πi 3 . The matrix K is an ∞ × 3m given by 1, 1). Then using the orthogonality Ω i 3 · Ω j 3 = 3δ i+j,3 and column operations, the matrix KA T can be expressed as The embedding σ K then gives an element U = Span C {B} ∈UGM with Multiplying B by z −g m j=1 (1 − (κ j z) 3 ) with g = g(S) = 2m, we find a basis of the subspace of meromorphic functions having pole only at z = 0, Now we set These variables then satisfy the following polynomial relations for (x, y 1 , y 2 ), . These relations are the minors of the following 2 × 3 matrix [24,12], (see (3.15)). These relations give a prime ideal of C[x, y 1 , y 2 ], and we have the coordinate ring, where P is the prime ideal defined by the relations above. Then one should note that the ring defines a singular space curve whose affine part is given by Spec(R). The ring (5.9) gives a subspace of the meromorphic functions on the singular space curves having pole only at ∞, i.e. R = Span C {Ã} with A = 1, x, x 2 , . . . , x m , y 1 , y 2 , x m+1 , xy 1 , xy 2 , x m+2 , x 2 y 1 , x 2 y 2 , . . . .
In Section 7, we will discuss a deformation of this singular space curve by setting where {β 3 j : j = 1, . . . , 2m} are distinct constants. The polynomial ringR given by C[x, y 1 , y 2 ]/P with defines a smooth space curveC (see Section 7). We also notẽ Thus each soliton solution considered here is associated with a coalescence of two moduli parameters of the smooth curveC i.e. β 3 j , β 3 m+j → κ 3 j . We remark that the singularity obtained by the coalescence is "not" an ordinary double point, but an ordinary 3-tuple point singularity as in [7].
Similar to Theorem 5.2, the Schur expansion of the τ -function can be obtained using the matrix KA T in (5.7). For example, consider S = 3, 7, 8 , (m = 2). We have where c µ (κ) is the Schur polynomial of (κ 3 1 , κ 3 2 ) associated to the Young diagram µ. Here the Young diagram µ in S µ increases by adding to the previous one, and the diagramμ in cμ increases by adding to the previous one (cf. Example 5.3). The corresponding Young diagram is λ(S) = (6, 4, 2, 2, 1, 1), i.e.

S
.
The matrix A is the 6 × 12 matrix, The base function (5.1) is given by ) n t n ). Then the Vandermonde matrix K is the ∞ × 12 matrix, The ∞ × 6 matrix KA T is then given by, after column operations, Then the Sato frame corresponding to the generalized soliton solution is given by We then set which defines a singular (plane) trigonal curve, That is, we have a coordinate ring, where the basisÃ of a set of meromorphic functions is given bỹ x, x 2 , y, x 3 , xy, x 4 , x 2 y, x 3 y, y 2 , . . . .
Note that the singular curve can be obtained by taking a triple point degeneration of a trigonal curve (called (3, 4)-curve), Thus, the plain curve for the generalized soliton solution associated with S = 3, 7 has two ordinary triple point singularities. This case was recently studied in [21].
The Schur expansion of the τ -function in this case is given by where each c µ (κ) is the Schur polynomial of (κ 3 1 , κ 3 2 ) associated with the Young diagram µ. Note that the Young diagrams λ in S λ (t) are obtained by adding or to the previous diagrams. Also note that each coefficient is a homogeneous polynomial but not a single Schur polynomial. The general formula of the coefficients cμ(κ) for κ = (κ 1 , . . . , κ m ) will be discussed elsewhere.

Proof of Theorem 5.1
We first summarize the notations and then give several steps to prove the theorem.
(1) The lmq-dimensional base for the exponential functions is given bŷ Each l-dimensional vector E(κ i ) is given by The Schur expansion ofÊ (j) (κ i ) is expressed bŷ whereK (j) is the ∞ × l matrix whose nth row is given by Row n K (j) (κ i ) = n + j j κ n i Ω n l for n = 0, 1, 2, . . .
Here Ω n l = (1, ω n l , ω 2n l , . . . , ω n(l−1) l ) with ω l = exp(2πi/l). The Schur expansion of the baseÊ is then expressed byÊ (2) The A-matrix is the N × mlq matrix given by mn i with n i = k(i − 1) + 1. CalculatingKA T , we havê The nth column of the ∞ × n q−j matrix [B (j) (κ i )] nq−j is given by (3) Consider the embedding σ K : A → K A T . Then the corresponding element of UGM can be expressed as follows: For each 0 ≤ j ≤ q − 1 and 1 ≤ n ≤ n q−j , we have which form the set of N elements in the base {φ −i (z) : i ∈ N 0 } for a point of UGM, i.e. we assign And the rest in the base {φ −i (z) : i ∈ N 0 } is given by {φ −i (z) = z −i : i ≥ N }. (4) Then we note that the set {φ −i (z) : 0 ≤ i ≤ N − 1} as a base of an N -dimensional vector space is equivalent to The other elements of the base are given by is a polynomial in z with F 0 (z) = 1 and deg(F i (z)) < i for i ≥ 1. More precisely, we have the following: First we have Let us now express a set of basis for these meromorphic sections: (i) For s j = 0, we have 0 ≤ j ≤ q − 1, and one can have an equivalent base for this case, that is, Here the number of elements is mq. (ii) For each s j with 1 ≤ s j ≤ k, we have 0 ≤ j ≤ q − 2. Then we write and an equivalent base can be expressed as Here the number of elements in the base is m(q − 1) for each 1 ≤ s j ≤ k. (iii) For each s j with k + 1 ≤ s j ≤ 2k, we have 0 ≤ j ≤ q − 3. Then an equivalent base is This continues to (q − 2)k + 1 ≤ s j ≤ (q − 1)k for just j = 0, and this gives whose number of elements is just m for each s 0 in (q − 2)k + 1 ≤ s 0 ≤ (q − 1)k.
Notice that the total number of elements in the bases in (i) through (iii) is just N = mq(1 + k(q − 1)/2). Then we have a base A(z) = A 1 (z) ∪ A 2 (z) for a subspace of C[z −1 ], where (5) Consider the index set for the minimal degrees of the polynomials in A = A 1 ∪ A 2 , i.e.
We now show that the set S forms a numerical semigroup of type l, lm + 1, lm + 2, . . . , lm + k . It is obvious that 0 ∈ S and |N 0 \ S| < ∞. The closeness under the addition is given as follows: Write p(r) and s(r) for 1 ≤ r ≤ q − 1 as Then each element can be expressed as which is in the span l, lm + 1, . . . , lm + k . For the element mql + i with i ≥ 0, we first write Then noting l − 1 ≤ qk, we have δ = µk + ν with 0 ≤ µ ≤ q and 0 ≤ ν ≤ k − 1, which leads to mql + i = µ(ml + k) + γl + m(q − µ)l + ν.
When k = 1, we have ν = 0 and mql + i is in the span l, lm + 1 . For the case with k > 1 and ν > 0, we have µ < q, hence Therefore the element mql + i is in the span l, lm + 1, . . . , lm + k for all i ≥ 0. We introduce the following variables, Then, the subspace spanned by the basis A can be expressed by a polynomial ring, where the prime ideal P is encoded in the set of 2 × 2 minors of the following 2 × (k + 1) matrix, Here the functions F (x) and G(x) are given by Note that the spectrum Spec(R) gives the affine part of a singular curve in the (k + 1)dimensional space of (x, y 1 , . . . , y k ). We also remark that this space curve is an irreducible component of the singular curve defined by . . , k (6) Let λ(S) be the Young diagram for the numerical semigroup S. Using the Binet-Cauchy formula, we can see that the τ -function has the expansion, wheret = {t n : n = ml for any m ∈ N} and c µ (κ) is a homogeneous symmetric polynomial of (κ l 1 , . . . , κ l m ), which is given by the N × N minor of the matrixKA T with the rows indexed by the diagram µ. Also note that the Young diagram associated with the matrix A is just λ(S) (see Theorem 2.2), i.e. A ∈ X λ(S) ⊂ Gr(N, M ). The pivot index of A is given by where s j for j = 0, 1, . . . , N − 1 are the first N ordered elements in S.

Deformation of the singular space curves for soliton solutions
Her we construct a smooth space curve for the numerical semigroup of type l, lm+1, . . . , l(m+1)−1 , i.e. k = l − 1. Then the smooth curve naturally degenerates to the singular curve associated with a soliton solution of the l-th generalized KdV hierarchy.
Let us recall that the singular curve for the soliton solution corresponding to this type has the local coordinates, Then we have the following algebraic relations (i.e. (6.1) with k = l − 1), with y 0 := F (x) = m j=1 (x − κ l j ) and y l := G(x) = xF (x). These relations can be expressed by the 2 × 2 minors of the following 2 × l matrix, (7.2) y 1 y 2 . . . y l−2 y l−1 G(x) F (x) y 1 . . . y l−3 y l−2 y l−1 .
Note that the relations in (7.1) give a prime ideal in C[x, y 1 , . . . , y l−1 ]. Inspired by the paper [12] (also see [24]), we consider the following deformation, where λ j for j = 1, 2, . . . 2m are all distinct. The singular limit is then given by coalescing two moduli parameters of the space curve, λ l j , λ l m+j −→ κ l j for j = 1, . . . , m.
One should note here that the singularity obtained by the above process is an ordinary l-tuple point singularity [7] (page 247). When l = 2, the hyperelliptic case, this is an ordinary double point singularity [19,28]. The smoothness of the curve can be shown as follows: We consider a commutative ring defined by where the prime ideal P is given by the relations in (7.1), (Hereafter we will omit the on the functions F (x) and G(x).) Then the affine part of the curve associated with the soliton solution is given by Spec(R). We now show that Spec(R) is non-singular.
Proposition 7.1. For every (x, y 1 , . . . , y l−1 ) satisfying all p i,j = 0, the following Jacobian U has rank l − 1, Here p i,j in each column of the matrix U are arranged in the following way, Proof. The l 2 × l Jacobian matrix has the following structure: (1) For the columns with p 1,j (2 ≤ j ≤ l), we have the (l − 1) × l matrix,  (2) For the columns with p i,j (i + 1 ≤ j ≤ l) and 2 ≤ i ≤ l − 3, we have the (l − i) × l matrix,  where the first nonzero entry in each row corresponds to ∂p i,j /∂y i−1 . (3) For the last three rows, we have  We consider the cases: (a) F (x) = 0, (b) G(x) = 0, and (c) F (x)G(x) = 0.
where the first nonzero entry in each row is at (i − 2)-th place, and y l = G. The total number of these rows is l−1 2 . This implies that the nullity of U T is l−1 2 , hence the rank of U is l 2 − l−1 2 = l − 1. This proves that the affine variety given by Spec(R) is a smooth curve.
A Riemann surface associated with the curve C can be obtained by adding one point ∞ to the affine smooth curve given by Spec(R). At ∞, we introduce the variables, Then we have a commutative ring, R = C[x, y 1 , . . . , y l−1 ]/P, where the prime ideal P is given by In the similar manner, we can prove that the affine curve given by Spec(R) is also smooth. Then the Riemann surface associated to the curve C is obtained by patching these affine curves. In terms of the notion of the Weierstrass semigroup [25], we have the following Corollary.
Finally, we remark that the spectrum Spec(R) is given by a projection of Spec( R) with the following ring R, We note a natural ring homomorphism R → R with This leads to a projection, We conclude the section with the following remark about the cases for the generalized soliton solutions, i.e. k < l − 1.
Remark 7.3. In the case of k = 1 in (6.1), we have just one relation, which is a singular plane curve. One can deform the curve with which then gives a smooth plane curve, called cyclic (l, lm + 1)-curve, The cases with l = 2 are just the hyperelliptic ones. The singular curve associated to the generalized soliton solution is obtained by coalescing l points of the (l, lm + 1)-curve, For the cases with 1 < k < l − 1, it seems in general that the singular curve is not smoothable in CP k+1 [24] [8]. One of a special case was found in [12] for l = 6, m = 2 and k = 4, i.e. the corresponding numerical semigroup is of type 6, 13, 14, 15, 16 . One can also show that the case of 4, 4m + 1, 4m + 2 (i.e. l = 4 and k = 2) has the following "smooth" deformation: The ring of the polynomials R is given by C[x, y 1 , y 2 ]/P where the prime ideal is given by where F 1 (x), F 2 (x) and G(x) are given by This smooth curve is an irreducible component of the intersection of the hypersurfaces given by We will discuss the deformation problem for the cases with 2 ≤ k ≤ l − 2 for l ≥ 5 in a future communication.

Spectral curves for soliton solutions
In this section, we identify the singular space curve given by (7.1) as a spectral curve of commuting differential operators in the Lax-Sato formulation of the KP hierarchy (see Appendix A, and also e.g. [9] for the formulation). This is an extension of the well-know theorem by Burchnall and Chaundy [4]. We here consider the cases for the soliton solutions corresponding to k = l − 1 and the generalized soliton solutions corresponding to k = l − 2. The general case with 1 ≤ k ≤ l − 3 will be left for the readers.
8.1. Spectral curve for the soliton solution (k = l − 1). We first recall that the τ -function of the soliton solution is given by τ (t) = |E(t)A T | in (2.6) where the m × lm matrices A is given by and E(t) whose base exponential functions (5.1) are given by Here E i (t, κ j ) is the exponential function given in (2.20), i.e.
Now we define the following time-operator, where σ i (κ) is the elementary symmetric polynomial of degree i in (κ l 1 , . . . , κ l m ), i.e.
Then we have the following proposition.
Proposition 8.1. The τ -function generated by (A, E(t)) above satisfies Proof. Since the τ -function depends only on the exponential functions E s (t, κ j ) for 1 ≤ s ≤ l, it is sufficient to show that each E s (t, κ j ) satisfies T α E s (t, κ j ) = 0 for all 1 ≤ α ≤ l − 1.

Direct computation shows
Since ω l l = 1, the term in the parenthesis in the equation becomes where we have used (8.1) with λ = κ l j . This proves the proposition. Remark 8.2. Note here that the equation T α τ (t) = 0 does not depend on the matrix A. This implies that any soliton solution of the l-th generalized KdV hierarchy satisfies Proposition 8.1. Also note that the number of the free parameters in the matrix A is m(l − 1), which is the genus of the corresponding numerical semigroup, i.e. g(S) = m(l − 1).
Since the solutions of the l-th generalized KdV hierarchy can be expressed by a single τ -function, they also satisfy T α v i = 0, where v i 's are from the l-th differential operator L l of (A.7) in Appendix A. Now we define the following differential operators of order ml + α in ∂, where B β is the differential operator of order β defined in (A.1), i.e. B β = (L β ) ≥0 . Then we have the following proposition. Again, note that the coefficients of the differential operators B n depend on t only through the τ -function. This implies that T α B β = 0 for any 1 ≤ α, β ≤ l − 1. This completes the proof.
In order to find the spectral curve given by the commuting differential operators {L l , L 1 , . . . , L l−1 }, we first note the following lemma. Proof. We first note that where we have used ∂ n φ = B n φ in (A.2). Then we write the wave function in the dressing form φ = W φ 0 as in (A.5) with (A.4). Noting again that the dressing operator depends on t only through the τ -function, we have This proves the lemma. Now using the coordinates in (3.13), i.e.
Then from Lemma 8.4, the following theorem is immediate (see Section 7).
Theorem 8.5. The eigenvalues of the commuting differential operators {L l , L 1 , . . . , L l−1 } satisfy the following relations, p i,j = y i y j−1 − y j y i−1 = 0 for 1 ≤ i < j ≤ l, where y 0 = F (x) = m j=1 (x − κ l j ) and y l = G(x) = xF (x). Remark 8.6. Theorem 8.5 can be considered as an extension of the well-known theorem by Burchnall and Chaundy [4] on the commuting pair of differential operators of positive order.    i (t, κ j ) = ∂ ∂κ j κ j E i (t, κ j ).
Since the matrix A and the base functions consist of m copies of the single case with different κ j 's, we here consider the case for m = 1. The general case for m > 1 follows in the similar manner as the case with m = 1. The τ -function for this case has the following expansion, with I i,ĵ = (i, l + 1, . . . , l + j, . . . , 2l), where l + j is the missing column index of the matrix A and the matrix E(t) is given by (1) (κ) ∂ 1 E(κ) . . . . . .
Then the exponential function E I i,ĵ (t) is expressed by the l × l Wronskian determinant, i (t, κ), E 1 (t, κ), . . . , E j (t, κ), . . . , E l (t, κ) , where the column of E j (t, κ) is removed from the matrix E(t). Then we have the following proposition.
where φ is called the wave function of the Lax pair (L, B n ). The compatibility among the equations ∂ n φ = B n φ, which is ∂ n ∂ m φ = ∂ m ∂ n φ, gives which is called the Zakharov-Shabat equations. The variable z ∈ C in (A.2) may be considered as a local coordinate at ∞ in the spectral space of L. Note that if the functions u i 's are all zero, then we have L = ∂ and B n = ∂ n and the wave function, denoted by φ 0 , is given by The wave function φ is then expressed in the dressing form, where the pseudo-differential operator W is called the dressing operator. Notice that all the functions u i 's in L can be determined by w j 's in W through L = W ∂W −1 .
Then, from the Lax equation, the dressing operator W satisfies (A.6) ∂ n (W ) = B n W − W ∂ n for n = 1, 2, · · · , which is sometimes called the Sato equation.
The l-th generalized KdV hierarchy is the l-reduction of the KP hierarchy defined by L l = (L l ) ≥0 , that is, the l-th power of L becomes a differential operator. This means that the functions u i 's are determined by l − 1 variables in L l in the form, (A.7) L l = ∂ l + v 2 ∂ l−2 + v 3 ∂ l−3 + · · · + v l−1 ∂ + v l .
Also note that those variables are determined by w i 's in W . From (A.1), the l-reduction gives the constraints, ∂ nl (L) = 0 for n = 1, 2, . . . , that is, all the variables v i 's do not depend on the times t nl . The original KdV hierarchy is given by the 2-reduction, and the solutions do not depend on the times t 2n .
A.2. N truncation and the τ -function. Here we explain the Wronskian formula of the τ -function. First note that a finite truncation of W with some positive integer N , given by is invariant under (A.6). We then consider the N -th order differential equation, For the time-evolution of the functions f i , we consider the following (diffusion) hierarchy, which gives the solution of the Sato equation (A.6). Then the solution of the KP equation can be expressed in terms of the τ -function by u(t) = 2u 2 (t) = 2∂ 1 w 1 (t) = 2∂ 2 1 ln τ (t).