Enumerative Geometry of Plane Curves

Lucia Caporaso

1. Spaces of Plane Curves the plane, ℙ2, of a nonzero homogeneous polynomial in Counting problems are among the most basic in math- three variables. A homogeneous polynomial of degree 푑 ematics. Enumerative geometry studies these problems in three variables, 푥0, 푥1, 푥2, has the form when they concern geometric entities, but its interaction 푗 퐺 = ∑ 푎 푥푖 푥 푥푘, (1) with other areas has been overwhelming over the past three 푑 푖,푗,푘 0 1 2 푖+푗+푘=푑 decades. In this paper we focus on algebraic plane curves 푖,푗,푘≥0 and highlight the interplay between enumerative issues where the coefficients 푎 are in ℂ. The set of all such and topics of a different type. 푖,푗,푘 polynomials is a complex vector space of dimension (푑+2) The classical ambient space for is the 2 complex , ℙ푟, viewed as a topological space and, by definition, two nonzero polynomials determine with the Zariski . The Zariski closed subsets are the same curve if and only if they are multiples of one defined as the the zero loci of a given collection of homoge- another. Therefore the set of all plane curves of degree 푑 neous polynomials in 푟+1 variables, with coefficients in ℂ. can be identified with the projective space of dimension 푑+2 푐푑 ≔ ( ) − 1 = 푑(푑 + 3)/2, These closed sets are called “algebraic varieties” when con- 2 sidered with the algebraic structure induced by the polyno- 푃 ≔ space of plane curves of degree 푑 = ℙ푐푑 . mials defining them. 푑 Plane curves are a simple, yet quite interesting, type of If 푑 is small, these spaces are well known. For 푑 = 1 algebraic variety. As sets, they are defined as the zeroes in we have the space of all lines, which is a ℙ2. For 푑 = 2 we have the space of all “conics,” a ℙ5. This is more in- Lucia Caporaso is professore ordinario di matematica at Università Roma Tre, teresting as there are three different types of conics: (a) Italy. Her email address is [email protected]. smooth conics, corresponding to irreducible polynomials; Communicated by Notices Associate Editor Daniel Krashen. (b) unions of two distinct lines, corresponding to the prod- For permission to reprint this article, please contact: uct of two polynomials of degree 1 with different zeroes; [email protected]. and (c) double lines, corresponding to the square of a poly- DOI: https://doi.org/10.1090/noti2094 nomial of degree 1. Notice that conics of type (c) form a

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 771 space of dimension 2 and conics of type (b) form a space system of 푐푑 homogeneous linear equations in 1 + 푐푑 un- 2 of dimension 4, as each of the two lines varies in ℙ . Since knowns (the 푎푖,푗,푘): the family of all conics has dimension 5, we see that most 퐺 (푝 ) = ⋯ = 퐺 (푝 ) = 0. conics are smooth or, with a suggestive terminology, “the 푑 1 푑 푐푑 general conic is smooth,” which is a shorthand for “the The solutions of this system form a vector space of dimen- set of smooth conics is dense and open in the space of all sion at least 1, with equality if and only if the equations conics.” are linearly independent, which will happen for general

The assortment of types of curves gets larger as the de- points 푝1, … , 푝푐푑 . Since a one-dimensional vector space of gree 푑 gets larger, but for any 푑 the general curve in 푃푑 is polynomials corresponds to a unique curve, we derive that smooth, i.e., given by a polynomial whose three partial there exists at least one curve through our fixed points, and derivatives have no common zeroes. the curve will be unique for a general choice of points. In So, smooth curves form a Zariski open dense subset in short 푃푑. This claim is an instance of a remarkable phenomenon 2 푐푑 =max {푛∶any 푛 points in ℙ lie in a curve of degree 푑} , in algebraic geometry. Indeed, let 푆푑 be the subset in 푃푑 parametrizing singular (i.e., nonsmooth) curves. By what and we solved our first, however easy, enumerative prob- lem by showing that the number of curves of degree 푑 pass- we said, 푆푑 is closed in 푃푑, hence the zero locus of some ing through 푐 general points is equal to 1. The phrase polynomials; therefore 푆푑 is an algebraic variety. More- 푑 “general points” means that the 푐 points vary in a dense over, as we shall see, the geometry of 푆푑 is all the more 푑 2 푐 interesting as it reflects some properties of the curves it open subset of (ℙ ) 푑 . parametrizes. The phenomenon we are witnessing is the Let us now focus on 푆푑, the space of singular plane fact that the sets parametrizing algebraic varieties of a cer- curves of degree 푑. It turns out that 푆푑 is a hypersur- tain type have themselves a natural structure of algebraic face in 푃푑, i.e., the set of zeroes of one polynomial, hence variety; they are usually called “moduli spaces” and are a dim 푆푑 = 푐푑−1. Arguing as before, the dimension of 푆푑 can central subject in current mathematics. be interpreted as the maximum number of points in the In this spirit, let us go back to plane curves and give an plane which are always contained in some singular curve interpretation to the dimension of the spaces of curves we of degree 푑. encountered so far. We introduced in (1) the general poly- For instance, four points always lie in some singular conic, and it is easy to describe which. If the four points nomial, 퐺푑, of degree 푑; now we consider the projective are general (i.e., no three are collinear), there are exactly space 푃푑 with homogeneous coordinates {푎푖,푗,푘, ∀푖, 푗, 푘 ≥ 2 six lines lines passing through two of them, and our con- 0 ∶ 푖+푗+푘 = 푑}, and the product 푃푑 ×ℙ . The polynomial ics are given by all possible pairs of them. This gives a total 퐺푑 is bihomogeneous of degree 1 in the 푎푖,푗,푘, and 푑 in the of three conics pictured in Figure 1. 푥푖. Therefore the locus where 퐺푑 vanishes is a well-defined 2 subset of 푃푑 × ℙ , and it is an algebraic variety which we denote by ℱ푑. We view ℱ푑 as a “universal family” of plane / O // curves of degree 푑. In fact we have the two projections, OOO / • •OO •/ o written 휋1 and 휋2,  OOO / oo  OO // o o • • OOO o/o • ℱ ⊂ 푃 × ℙ2 (2)  OO ooo / 푑 t 푑 JJ • • • • OOO oo • •/ tt JJ  ooo // 휋1tt J휋2  oo / tt JJ  oo tt JJ ztt J% 2 푃푑 ℙ Figure 1. The three singular conics through four points. and the restriction of 휋1 to ℱ푑 expresses it as a family of plane curves: the preimage in ℱ푑 of a point, [푋] ∈ 푃푑, If three of the fixed points are collinear, we take all parametrizing a curve, 푋 ⊂ ℙ2, is isomorphic to 푋, and conics given by the union of the line through the three 2 it is mapped to 푋 by the projection, 휋2, to ℙ . points with any line through the fourth point; since the The fact that 푃푑 has dimension 푐푑 = 푑(푑 + 3)/2 tells set of lines through a point has dimension one, we get a 2 us that if we fix 푐푑 points in ℙ there will exist some one-dimensional space of conics. If the four points are curve of degree 푑 passing through them, and the curve collinear, we have the two-dimensional space of conics will be unique for a general choice of points. In fact, fix given by the union of the line through the points with an 2 푝1, … , 푝푐푑 ∈ ℙ ; a curve passes through 푝푖 if the poly- arbitrary line. nomial defining it vanishes at 푝푖. Therefore the curves Summarizing, if (and only if) the four points are gen- passing through our points are determined by imposing eral (i.e., no three are collinear), there exist finitely many 퐺푑(푝푖) = 0 for all 푖 = 1, … , 푐푑. This gives the following singular conics through them, and the number of such

772 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 conics is always three, regardless of the choice of the four by the solutions of the following system, where 퐹푥푖 is the points. partial derivative with respect to 푥푖, As easy as this is for conics, things get more complicated 퐹푥0 = 퐹푥1 = 퐹푥2 = 0. (4) already for 푑 = 3. Here dim 푆3 = 8, and counting the

“cubics” through eight points is much harder. Since 퐹 is bihomogeneous of bidegree (1, 푑), each 퐹푥푖 is The key is to give this number a different interpretation1 bihomogenous of bidegree (1, 푑 − 1) and corresponds to a 1 2 and identify it with another invariant of 푆푑, its degree as a hypersurface in ℙ × ℙ of the same bidegree. The number subvariety of 푃푑. of solutions of the system (4) is thus the number of points of intersection in ℙ1 ×ℙ2 of three hypersurfaces of bidegree 2. The Degree of the Severi Variety (1, 푑 − 1). The degree of a subvariety in projective space is the num- We do know how to compute this number because we ber of points of intersection with as many generically cho- can compute intersections in projective space. ∗ 푟 sen hyperplanes as its dimension. In 푃푑 there are hyper- We write 퐻 (ℙ ) for the ring with ℤ- planes with a special geometric meaning, parametrizing coefficients of the projective space ℙ푟, whose cup product curves passing through a fixed point. Indeed, let 푝 be a can be interpreted as the intersection product. As a ring, 2 푝 ∗ 푟 푟+1 point in ℙ , and let 퐻 be the locus in 푃푑 of curves through 퐻 (ℙ ) is isomorphic to ℤ[푥]/(푥 ), where 푥 is identified 2 푟 푝: with the cohomology class, ℎ푟 ∈ 퐻 (ℙ ), corresponding to 푝 푘 2푘 푟 퐻 = {[푋] ∈ 푃푑 ∶ 푝 ∈ 푋}. (3) a hyperplane. Hence 푥 ∈ 퐻 (ℙ ) corresponds to a linear 푝 subspace of complex dimension 푟 − 푘 and real codimen- Thus 퐻 is the zero locus in 푃푑 of the homogeneous linear 푟 푝 sion 2푘. The intersection product in ℙ depends only on polynomial 퐺푑(푝), hence 퐻 is a hyperplane. Therefore, the cohomology class, and the degree of the intersection deg 푆푑, the degree of 푆푑, is the number of singular curves of 푟 hypersurfaces is the product of their degrees; in alge- passing through dim 푆푑 general points, as claimed. Recall- braic geometry, this is B´ezout’s theorem. This degree is the ing that dim 푆푑 = 푐푑 − 1, we want to solve the following. appropriate count for the number of points of intersection Problem 1. Compute the number of singular plane curves of the 푟 hypersurfaces. of degree 푑 passing through 푐푑 − 1 general points. Equiva- One usually identifies zero-dimensional classes, like the 2 lently, compute the degree of 푆푑. class of the intersection of two curves in ℙ , with their de- gree. This amounts to identifying the top cohomology Since 푆 is a hypersurface, its degree is equal to the num- 푑 group, 퐻2푟(ℙ푟), with ℤ so that the class of a point corre- ber of points of intersection with a general line. So, we fix sponds to 1. a general line, 퐿, in 푃 and notice that 퐿 corresponds to a 푑 The generator, ℎ , for ℙ1 is the dual of a point, and the family of curves of degree 푑, a so-called “pencil of curves.” 1 generator, ℎ , for ℙ2 is the dual of a line, with ℎ2 = 0 and ℎ2 More precisely, from diagram (2) we restrict the projection 2 1 2 equal to the class of a point; with the above identification, ℱ푑 → 푃푑 over 퐿 to get a map 2 we write ℎ2 = 1. 1 2 휙 ∶ 풳 ⟶ 퐿 ⊂ 푃푑 What about ℙ ×ℙ ? It satisfies a Künneth type formula, so that its cohomology ring is generated by the pullbacks whose fiber over every point, ℓ ∈ 퐿, is the plane curve of the generators of the two factors. Let us denote by 픥 the of degree 푑 corresponding to the curve parametrized by ℓ. 푖 pullback of ℎ for 푖 = 1, 2. The word “pencil” indicates that the base of the family, 퐿, 푖 By what we said, the number of solutions of the system is a line. (4) is the degree of the triple intersection of the class 픥 + We identify 퐿 with ℙ1 and denote by 푡 , 푡 its homoge- 1 0 1 (푑 − 1)픥 . The following basic relations are easily seen to neous coordinates. Then our pencil is given by the zeroes 2 hold (again identifying the top cohomology group with in ℙ1 × ℙ2 of a polynomial 푡0,푡1 푥0,푥1,푥2 ℤ): 3 2 3 2 퐹(푡0, 푡1; 푥0, 푥1, 푥2), 픥1 = 픥1픥2 = 픥2 = 0, 픥1픥2 = 1.

bihomogeneous of degree 1 in 푡0, 푡1 and 푑 in 푥0, 푥1, 푥2, so Hence 1 2 3 2 that 풳 ⊂ ℙ × ℙ is set of zeroes of 퐹. Since 퐿 is a gen- (픥1 + (푑 − 1)픥2) = 3(푑 − 1) . 푃 푆 eral line in 푑, it intersects 푑 transversally in finitely many Therefore the number of singularities in the fibers of 휙 is 퐿 휙 points. These are the points of such that the fiber of equal to 3(푑 − 1)2. is singular, and our goal is to count them. We first count By the generality of the line 퐿, every singular fiber has 휙 the singular points of the fibers of , which are determined exactly one singular point. Hence the number of singular fibers of 휙 is 3(푑 − 1)2, and hence 1“Mathematics is the art of giving the same name to different things.” 2 —H. Poincar´e deg 푆푑 = 3(푑 − 1) (5)

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 773 is the answer to Problem 1. This confirms that there are three singular conics through four general points, and it tells us, for example, that there are twelve singular cubics passing through eight general points. Notice that, differ- ently from what happens with conics, if the eight points contract loop are general (no three are collinear, no six on a conic), each desingularize node of these cubics will be irreducible, i.e., not the the union of a line and a conic. In answering Problem 1 we mentioned that the general curve in 푆푑 has exactly one singular point. Moreover, this point is a “node,” the simplest type of singularity a curve Figure 2. A singular specialization and its desingularization. can have, whose analytic local equation has the form 푥2 = 푦2. The geometric genus of an irreducible singular curve is We now consider curves with more singular points. We defined as the genus of its desingularization. denote by 푆 the Severi variety of plane irreducible curves 푑,훿 The arithmetic genus can be thought of as the total en- of degree 푑 with at least 훿 nodes; see [14]. More precisely, ergy of the curve, with the geometric genus being the po- 푆 is defined as the closure in 푃 of the locus of irreducible 푑,훿 푑 tential energy. Just like the total energy of a system remains curves of degree 푑 with 훿 nodes: constant, so does the arithmetic genus in a family of curves of fixed degree. On the other hand the potential energy 푆푑,훿 ≔ {[푋] ∈ 푃푑 ∶ 푋 irreducible with 훿 nodes}. can be converted, all or part of it, to a “less useful” energy, If 훿 = 0, then 푆푑,0 = 푃푑; if 훿 = 1 and 푑 ≥ 3, we have and indeed the geometric genus of a curve can decrease in 푆푑,1 = 푆푑. a specialization, but never increase. In particular, a family It is clear that for 푆푑,훿 to be nonempty, 훿 cannot be too of curves of genus 0 specializes to a curve of genus 0, all of big, for example it is easy to see that 훿 must be less than whose irreducible components must have genus 0. (푑 − 1)2/2. Indeed, suppose we have an irreducible curve, What about positive genus? Consider a family of 푋, of degree 푑 ≥ 3 with at least (푑 − 1)2/2 nodes. Through smooth curves of degree 푑 ≥ 3 specializing to an irre- these nodes there certainly passes a curve, 푌, of degree 푑−2, ducible curve with 훿 nodes. The underlying family of topo- 2 because 푐푑−2 > (푑 − 1) /2. Hence the degree of the inter- logical surfaces has, as general fiber, a surface of genus section of 푋 and 푌 is at least 2(푑 − 1)2/2 = (푑 − 1)2, but 푔 = (푑−1), hence with 푔 handles; see Figure 2 for a pic- 2 this contradicts B´ezout’s theorem, according to which the ture with 푔 = 1 and 훿 = 1. In this family every node of degree of the intersection of 푋 and 푌 is 푑(푑 − 2). A more the specialization is generated by the contraction of a loop 푑−1 refined analysis gives ( ) as a sharp upper bound on 훿, around a handle of the general fiber. The surface underly- 2 and we have ing the special curve is no longer a topological manifold at the point where the loop got contracted, where the surface 푑−1 Fact 2.1. If 훿 > ( ), then 푆푑,훿 is empty. Assume 훿 ≤ 2 looks locally like two disks with centers identified. Sep- (푑−1), then arating the two disks desingularizes the curve so that the 2 underlying surface has one fewer handle, hence its genus (a) 푆푑,훿 is irreducible of dimension 푐푑 − 훿; goes down by 1. (b) 푆푑,훿 is smooth at points parametrizing irreducible This was an informal explanation for the fact that the curves with exactly 훿 nodes, and the locus of such geometric genus of an irreducible curve of degree 푑 with 훿 points is open and dense in 푆 . 푑−1 푑−1 푑,훿 nodes is equal to ( )−훿; hence if 훿 > ( ) there exist no 2 2 The irreducibility of 푆푑,훿 is proved in [9]. The num- such curves. We refer to [1] and [2] for the general theory ber (푑−1) is the arithmetic genus of a plane curve of degree of curves. 2 Recapitulating, 푆 can be defined as the the closure in 푑. There are two types of genus for a curve: the geometric 푑,훿 푃 of the locus of irreducible curves with geometric genus genus and the arithmetic genus, which coincide if the curve 푑 푔 = (푑−1) − 훿. A simple calculation gives a different is smooth and irreducible. 2 The genus of a smooth curve is the topological genus expression for its dimension: of the real surface underlying the curve. For example, a smooth plane curve of degree 1 or 2 has genus 0 and its dim 푆푑,훿 = 3푑 + 푔 − 1. underlying real surface is the sphere, 푆2. In degree 3, the genus is 1 and the surface underlying a smooth cubic is a The basic enumerative problem, generalizing Problem 1, torus. is the following.

774 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 Problem 2. Compute the number of irreducible plane curves of degree 푑 and genus 푔 passing through 3푑 + 푔 − 1 general points.

푑−1 Or, compute the degree of 푆푑,훿. If 훿 = 0, then 푔 = ( ), 푋1 푋1 2 and the answer is 1. If 훿 = 1, then 푔 = (푑−1) − 1, and we 2 know the answer is 3(푑 − 1)2. For 훿 ≤ 8 the answer, given 푋2 푋2 in [10], is again a polynomial in 푑; see also [5] and [7]. For Figure 3. Reducible specializations of rational quartics. bigger 훿 the first general solutions were discovered as recur- sive, rather than closed, formulas, as we shall illustrate in we used to compute the degree of 푆푑, but with an oppo- the rest of the paper. site point of view. In the previous case the unknown was 3. Recursive Enumeration for Rational Curves the number of special curves. Now the number of special curves will be easy to compute and will be used to deter- A rational curve is an irreducible curve of geometric genus mine the unknown, 푁푑. 0. By what we said earlier, a family of rational curves can To get our one-dimensional family we intersect 푅푑 with specialize only to a curve all of whose irreducible compo- general hyperplanes until we get a curve. Since intersect- nents are rational. This makes the enumeration problem ing with a general hyperplane decreases the dimension by in genus 0 self-contained and solvable by a recursive for- 1, we need to intersect with 푟푑 − 1 hyperplanes. We use mula, which expresses the degree of the Severi variety in 푝 hyperplanes of type 퐻 , defined in (3). So, fix 푞1, … , 푞푟 −1 degree 푑 and genus 0 in terms of the degrees of the Severi 푑 general points in ℙ2 and set varieties in lower degrees and genus, again, 0. 푞 퐶 = 푅 ∩ 퐻푞1 ∩ ⋯ ∩ 퐻 푟푑−1 . We denote by 푅푑 the Severi variety of rational curves of 푑 degree 푑: Now 퐶 is the curve in 푃푑 parametrizing the family of rational curves of degree 푑 through the basepoints 푅푑 ≔ 푆푑,(푑−1), 푟푑 ≔ dim 푅푑 = 3푑 − 1, 푁푑 ≔ deg 푅푑. 2 푞1, … , 푞푟푑−1, which we write as follows (6) 퐶 × ℙ2 ⊃ 풳 ⟶ 퐶. Of course, 푁1 = 1. In case 푔 = 0 the answer to Problem 2 is the following. The basic idea is that our degree, 푁푑, is equal to the num- ber of points of intersection between 퐶 and one more gen- Theorem 3.1 (Kontsevich’s formula). For 푑 ≥ 2, eral hyperplane 퐻푝, as this is equal to the number of curves 푅 푞 , … , 푞 , 푝 3푑 − 4 3푑 − 4 in 푑 passing through 1 푟푑−1 . To put this idea to 푁 = ∑ 푁 푁 푑 푑 푑 푑 − 푑2 . 푑 푑1 푑2 1 2 [( ) 1 2 ( ) 2] work, we need to study the geometry of the family 풳 → 퐶. 3푑1 − 2 3푑1 − 3 푑1+푑2=푑 By Fact 2.1 and Bertini’s theorems, 퐶 is irreducible and As explained in [11], this formula was discovered in the subset of points parametrizing irreducible curves with (푑−1) nodes and no other singularities is open, dense, and a rather different context, and it came as a beautiful sur- 2 prise. While establishing the mathematical foundations contained in the smooth locus of 퐶. Let us look at the re- for Gromov–Witten theory (a theory largely inspired by ducible curves parametrized by 퐶. There are finitely many ideas from physics), Kontsevich and Manin gave an ax- of them and, by what we said earlier, they must be unions iomatic construction of the Gromov–Witten invariants of two rational curves of smaller degrees. and of the ring, a generalization of Example 3.2. Let 푑 = 4, hence 훿 = 3 and 푟 = 11. Our the classical cohomology ring of a projective algebraic va- 4 퐶 parametrizes quartics with three nodes passing through riety. For ℙ2 the quantum product on the quantum coho- ten points. The reducible curves parametrized by 퐶, writ- mology ring was defined using our numbers 푁 , which ap- 푑 ten 푋 ∪ 푋 , are of two types, drawn in Figure 3. First type: peared as Gromov–Witten invariants. The above formula 1 2 for any partition of the basepoints into two subsets, 퐵 and was found as the condition characterizing the associativity 1 퐵 , of five points, let 푋 be the conic through 퐵 for 푖 = 1, 2 of the quantum product. 2 푖 푖 (drawn on the left in the picture). Second type: for any The proof we shall illustrate was not among the first to partition of the basepoints into a subset, 퐵 , of cardinal- be given (for which we refer to [11] and [12], or to [13]), 1 ity 2 and a subset, 퐵 , of cardinality 8, let 푋 be the line but is close in spirit to our answer to Problem 1. 2 1 through 퐵 , and let 푋 be one of the twelve nodal cubics The shape of the formula indicates that we should use 1 2 through 퐵 . splittings of the curve into a union of two components of 2 degrees 푑1 and 푑2 with 푑 = 푑1 + 푑2. We will do that via We use the following notation: 푋1 ∪ 푋2 denotes a a one-dimensional family of curves similar to the pencil reducible curve of our family, with 푋1 containing the

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 775 basepoint 푞1. The degree of 푋1 will be 푑1, and 푋2 has de- gree 푑2 = 푑−푑1. We say that such curves are of type (푑1, 푑2). Let us count them; we have Desingularize 풵 풵′ 푟푑1 + 푟푑2 = 푟푑 − 1, which is the number of basepoints of our family. There- fore for every partition of the basepoints into two subsets, 퐵 and 퐵 , of respective cardinalities, 푟 and 푟 , there exist 1 2 푑1 푑2 Figure 4. Reducible specialization and desingularization. 푁푑푖 rational curves of degree 푑푖 passing through 퐵푖. Since we are assuming that the basepoint 푞1 lies on 푋1, the num- that 휙 ∶ 풵′ → 푈 is a family whose fibers away from 푢 is a ber of curves of type (푑1, 푑2) is equal to 푁푑 푁푑 times the 0 1 2 smooth rational curve. number of partitions of the basepoints 푞2, … , 푞푟 −1 into 푑 The curve 푋 ∪푋 will be desingularized at all nodes but two subsets of cardinalities 푟푑 − 1 and 푟푑 , hence equal to 1 2 1 2 the one which is smoothed in 풵 (i.e., the node which is not 푟 − 2 a limit of nodes, marked by a circle in Figure 4), hence the 푁 푁 ( 푑 ). (7) 푑1 푑2 fiber of 휙 over 푢 is reducible with exactly one node. If, by 푟푑1 − 1 0 푋 ∩ 푋 Now, is 퐶 singular at such special points? As 퐶 is a contradiction, all points in 1 2 were limits of general nodes, then they will all be desingularized when passing general linear section of the Severi variety 푅푑, its singular- ′ to 풵 , and the fiber of 휙 over 푢0 will be disconnected. But ity at any point reflects the local geometry of 푅푑, which depends on the singularities of the corresponding plane this contradicts the connectedness principle, as the fibers 휙 푢 curve. Let us count the singular points of a curve of type of away from 0 are all connected. 푅 푋 (푑 , 푑 ). This amounts to counting the nodes of each com- In conclusion, the local geometry of 푑 at a curve, , 1 2 (푑 , 푑 ) ponent and the 푑 푑 nodes in which the two components of type 1 2 reflects that in a general deformation there 1 2 푋 intersect, which gives a total of is exactly one node of which gets smoothed, so that 푅푑 is the intersection of smooth branches, each of which 푑 − 1 푑 − 1 푑 − 1 ( 1 ) + ( 2 ) + 푑 푑 = ( ) + 1. corresponds to the node of 푋 which gets smoothed along 2 2 1 2 2 that branch. By what we said, only the 푑1푑2 “intersection” Now, when rational curves of degree 푑 specialize to a nodes can be smoothed, and the fact that all such nodes 푑−1 curve of type (푑1, 푑2) each of the ( ) nodes of the gen- are actually smoothable follows by a symmetry argument. 2 eral fiber specializes to a node of the special fiber. Bythe Concluding, 푅푑 is, locally at 푋, the transverse intersection above computation, there is exactly one node of the spe- of 푑1푑2 smooth branches. Hence the curve 퐶 has an ordi- cialization which is not the limit of a “general” node. In nary 푑1푑2-fold point and, under 퐵 → 퐶, the preimage of a other words, the special curve has exactly one node that point of type (푑1, 푑2) is made of 푑1푑2 distinct points. gets “smoothed.” But can all the nodes of the special curve We want to compute 푁푑 using, as for Problem 1, in- tersection theory. We start from the family of curves be smoothed in this way? No, only the 푑1푑2 nodes lying in the intersection of the two components can. parametrized by 퐶 and let 퐵 → 퐶 be its desingularization. We pull back to 퐵 the original family 풳 → 퐶 but, as we To see why, suppose we have a curve, 푋 = 푋1 ∪ 푋2, of noted above, the so-obtained surface is singular along the type (푑1, 푑2) occurring as the specialization of a family as parametrized above by a smooth curve, 푈. Write 풵 → 푈 nodes of its irreducible fibers, hence we replace it by its desingularization, 풴. We have a commutative diagram for this family, and let 푢0 ∈ 푈 be the point parametrizing 푋. Up to shrinking 푈 near 푢0, we can assume that all fibers 푑−1 휋 away from 푢0 are irreducible with ( ) nodes. The surface 2  ( / / 2 / 2 풵 is necessarily singular along the nodes of the irreducible 풴 풳 ℙ × 푅푑 ℙ fibers, hence it has (푑−1) singular curves, which we resolve 2 휑 by desingularizing 풵. In Figure 4 we have the example of    /  / rational quartics specializing to a reducible curve (see also 퐵 퐶 푅푑 Figure 3 and Example 3.2); the dotted red curves represent the singular curves of 풵. Denote by 풵′ the desingulariza- where 휑 is a family of rational curves with finitely many tion of 풵 so that we have a chain of maps reducible fibers. Our 풴 is a ruled surface, birational to ℙ1 × 퐵, and its 휙 ∶ 풵′ ⟶ 풵 ⟶ 푈 intersection product is something we can handle. First, we whose composition is a new family of curves. This opera- need a section of 휑. Every basepoint of the family deter- tion has the effect of desingularizing the general fibers, so mines such a section; let 푄 be the section corresponding

776 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 to 푞1. Thus 푄 is the curve in 풴 intersecting each fiber of 휑 intersection ring of 풴, we can write in one point and such that 휋(푄) = 푞 . 1 휋∗ℎ = 푐 푄 + 푐 푌 + ∑ 푐 푍 (9) We denote by 푇 ⊂ 퐵 the set of points over which the 2 푄 푌 푏 푏,2 푏∈푇 fiber of 휑 is reducible. For every 푏 ∈ 푇, we denote its fiber for some coefficients 푐 , 푐 , 푐 . We can compute these co- by 푍푏,1 ∪ 푍푏,2 with the convention that the map 휋 sends 푄 푌 푏 efficients by intersecting both sides of (9) with the three 푍푏,푖 to a plane curve of degree 푑푖, and the curve of degree 푑1 types of generators. We have contains 푞1; so 푍푏,2 does not intersect 푄. We write 푇(푑1, 푑2) for the set of points parametrizing a curve of type (푑 , 푑 ), ∗ ∗ ∗ 1 2 휋 ℎ2 ⋅ 푄 = 0, 휋 ℎ2 ⋅ 푌 = 푑, 휋 ℎ2 ⋅ 푍푏,2 = 푑2 and 푡(푑1, 푑2) for its cardinality. We computed in (7) the number of points in 퐶 by construction. These give linear relations which, with the intersection table, enable us to determine the coeffi- parametrizing curves of type (푑1, 푑2). Over each such point cients and obtain there are 푑1푑2 points of 퐵, hence

푟푑 − 2 ∗ 2 푡(푑 , 푑 ) = 푑 푑 푁 푁 ( ). (8) 휋 ℎ2 = 푑푄 − (푑푄 )푌 − ∑ ( ∑ 푑2푍푏,2) . 1 2 1 2 푑1 푑2 푟 − 1 푑1 푑1+푑2=푑 푏∈푇(푑1,푑2) ∗ 2 As 풴 is a ruled surface, its intersection ring is generated Now, since (휋 ℎ2) = 푁푑, we have by the curves 2 2 2 푁푑 = −푑 푄 + ∑ (푑2푍푏,2) {푄, 푌, 푍푏,2 ∀푏 ∈ 푇}, 푑1+푑2=푑 and we will use the same symbols for curves and their co- 푏∈푇(푑1,푑2) homology classes. We have the obvious relations, for every 푑2 = ∑ [ 푠(푑 , 푑 ) − 푑2푡(푑 , 푑 )] . ∀푏 ∈ 푇, 2 1 2 2 1 2 푑1+푑2=푑 2 2 푄 ⋅ 푌 =1, 푄 ⋅ 푍푏,2 =0, 푌 ⋅ 푍푏,2 =0, 푌 =0, 푍푏,2 =−1, Hence 2 identifying, as before, the top cohomology group of 풴 with 푑 푟푑 − 3 2 푟푑 − 2 푁푑 = ∑ 푁푑1 푁푑2 푑1푑2 [ ( ) − 푑2( )] . 2 2 푟푑 − 1 푟푑 − 1 ℤ. To complete the intersection table, we need 푄 . We 푑1+푑2=푑 1 1 compute it using a second section from the basepoints, so To see that this formula gives Theorem 3.1, set 푛 = 푟 −3 = let 푄′ be the section corresponding to, say, 푞 . We have 푑 푟푑−1 3푑 − 4 and 푘 = 푟 − 1 = 3푑 − 2. For the term in square 푄 ⋅ 푄′ = 0 and (푄′)2 = 푄2, therefore 푄2 = (푄 − 푄′)2/2. 푑1 1 brackets we have Now, the intersection numbers of 푄 and 푄′ differ only 푑2 푛 푛 + 1 on the generators 푍푏,2 for which the basepoint 푞푟푑−1 lies on 2 ′ ( ) − 푑2( ) 휋(푍푏,2), in which case we have 푄 ⋅푍푏,2 = 1. Write 푆 ⊂ 푇 for 2 푘 푘 the set of such points, so that 푄 − 푄′ and ∑ 푍 have 푑2 + 푑2 푛 푛 푛 푛 푏∈푆 푏,2 = 1 2 ( ) + 푑 푑 ( ) − 푑2( ) − 푑2( ) the same class in the intersection ring of 풴. Hence 2 푘 1 2 푘 2 푘 2 푘 − 1 푛 푛 푑2 − 푑2 푛 1 1 = 푑 푑 ( ) − 푑2( ) − 2 1 ( ). 푄2 = (푄 − 푄′)2 = (∑ 푍 )2 1 2 푘 2 푘 − 1 2 푘 2 2 푏,2 푏∈푆 Summing up, for 푑1 +푑2 = 푑 the third summand vanishes, #푆 1 = − = − ∑ 푠(푑 , 푑 ), and we are done. 2 2 1 2 푑1+푑2=푑 For details we refer to [3], where this technique is used to obtain other recursions enumerating rational curves on where 푠(푑1, 푑2) is the number of points in 푆 of curves of rational surfaces. type (푑1, 푑2). Arguing similarly as for 푡(푑1, 푑2), we obtain 푟 − 3 4. Curves of Positive Genus 푠(푑 , 푑 ) = 푑 푑 푁 푁 ( 푑 ). 1 2 1 2 푑1 푑2 We now look at Problem 2 for 푔 > 0. A complete answer is 푟푑1 − 1 provided by means of a recursion, the precise description Now, the preimage of a general point, 푝, in ℙ2 un- of which would require too many new technical details. der the map 휋 is the set of curves of our family passing We thus limit ourselves to illustrating the main idea and through 푝, whose cardinality is, of course, deg 휋. Hence the comparison with the previous formulas. deg 휋 is the number of rational curves of degree 푑 passing If we consider, as we did for 푔 = 0, the family of curves through 푞1, … , 푞푟푑−1, 푝, that is our unknown, 푁푑. On the of degree 푑 with 훿 nodes passing through a number of 2 other hand, let ℎ2 be the class of a line in ℙ , then deg 휋 points equal to dim 푆푑,훿 −1, we will get a one-dimensional ∗ 2 is equal to (휋 ℎ2) , an on 풴. Hence family which, in contrast with the case of rational curves, ∗ 2 computing 푁푑 is the same as computing (휋 ℎ2) . In the will parametrize no reducible curve, in general.

JUNE/JULY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 777 Example 4.1. Let 푑 = 5 and 훿 = 2 so that the geometric Example 4.2. Consider the generalized Severi variety with genus is 4 and dim 푆5,2 = 18. Let 퐶 be the linear section 푑 = 4 and 훿 = 3, defined as the closure in 푃4 of the of 푆5,2 parametrizing all quintics with two nodes passing set of quartics with three nodes (so 훼 = (0, … , 0) and through seventeen points. There exists no reducible quin- 훽 = (4, 0, … , 0)). This variety is the union of two irreducible tic passing through seventeen general points, as 푐1 + 푐4 = components, both of dimension 11. One component is 2+14 = 16 and 푐2 +푐3 = 5+9 = 14. Hence 퐶 parametrizes 푆4,3, whose general point parametrizes irreducible ratio- only irreducible curves. nal quartics (the same considered in Example 3.2). The second component parametrizes reducible quartics given To set up a recursive approach we need some further by the union of a line and a cubic. Since 푐1 = 2 and 푐3 = 9, constraint to force reducible curves to appear. Our method this component has dimension 11. is to impose that the basepoints lie all on a fixed line in the plane. Since an irreducible curve cannot meet a line The degree of the generalized Severi variety is computed in more points than its degree, imposing a high enough in [4] by a recursive formula from which the (more intri- number of basepoints on a line will certainly cause the oc- cate) recursion for the degree of 푆푑,훿(훼, 훽) follows. The currence of reducible curves. proof is based on a thorough analysis of the geometry For this idea to work, we must impose one basepoint of the Severi variety, a subject of its own interest. More at a time, in order to be able to tell at which step re- recently, a different proof of the same formula has been ducible curves appear, and to be able to describe them. given in [6] using tropical geometry. This will occur at various steps, and we will have to han- In [15] an approach similar to [4] is used to enumer- dle sections of our Severi variety having arbitrary dimen- ate curves of any genus in general rational surfaces. As for sion. Therefore we cannot limit our study to families over the enumerative geometry of curves on different surfaces, a one-dimensional base, as we did for the earlier formulas. some results are known and some interesting conjectures More precisely, the procedure starts by fixing fix a line, 퐿, are under investigation by means of diverse techniques. As 2 we cannot, for lack of space, give here an exhaustive list of in ℙ and a certain number of general points 푞1, 푞2,…, on 퐿. Now we intersect the Severi variety with the hyperplane references, we refer to [8] and to its bibliography. 퐻푞1 퐻푞2 , then with , and so on, by keeping track of how References the intersection behaves at every step. After a certain num- [1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Ge- ber of steps the intersection splits into irreducible compo- ometry of algebraic curves. Vol. I, Grundlehren der Mathema- nents, some of which parametrize curves of type 퐿 ∪ 푋, so tischen Wissenschaften [Fundamental Principles of Math- that 푋 has lower degree and the recursion kicks in. ematical Sciences], vol. 267, Springer-Verlag, New York, The price of this recursive method is that we must con- 1985. MR770932 sider a new version of Severi variety, where the novelty is [2] Enrico Arbarello, Maurizio Cornalba, and Phillip A. Grif- in the prescription of certain orders of contact with 퐿 at fiths, Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Princi- some of the basepoints 푞푖, and at arbitrary points. In other words, we need to introduce the Severi variety parametriz- ples of Mathematical Sciences], vol. 268, Springer, Heidel- berg, 2011. With a contribution by Joseph Daniel Harris. ing plane curves of fixed (degree, genus, and) intersection MR2807457 퐿 profile with the line . [3] Lucia Caporaso and Joe Harris, Enumerating rational We set 훼 = (푎1, … , 푎푛) and 훽 = (푏1, … , 푏푚), with 푎푖 and curves: the rational fibration method, Compositio Math. 113 푏푖 nonnegative integers such that ∑ 푖푎푖 + ∑ 푖푏푖 = 푑. On (1998), no. 2, 209–236, DOI 10.1023/A:1000446404010. the line 퐿 we fix ∑ 푎푖 general points, {푞푖,푗} with 푖 = 1, … , 푛 MR1639187 and 푗 = 1, … , 푎푖. We define 푆푑,훿(훼, 훽) to be the closure in [4] Lucia Caporaso and Joe Harris, Counting plane curves of 푃푑 of the locus of irreducible curves with 훿 nodes, having any genus, Invent. Math. 131 (1998), no. 2, 345–392, DOI 10.1007/s002220050208. MR1608583 (a) a point of contact of order 푖 with 퐿 at 푞푖,푗 for every [5] P. Di Francesco and C. Itzykson, Quantum intersection 푖 = 1, … , 푛 and 푗 = 1, … , 푎푖; rings, The of curves (Texel Island, 1994), (b) 푏푖 points of contact of order 푖 with 퐿 for every 푖 = Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1, … , 푚, different from those in (a). 1995, pp. 81–148, DOI 10.1007/978-1-4612-4264-2_4. MR1363054 For 훼 = (0, … , 0) and 훽 = (푑, 0, … , 0) we recover the clas- [6] Andreas Gathmann and Hannah Markwig, The Caporaso- sical Severi variety 푆푑,훿. Harris formula and plane relative Gromov–Witten invariants in One final point: in this setup it is natural to drop the tropical geometry, Math. Ann. 338 (2007), no. 4, 845–868, condition that the general curve be irreducible. So, next to DOI 10.1007/s00208-007-0092-4. MR2317753 푆푑,훿(훼, 훽) we consider the generalized Severi variety, defined as before but omitting the irreducibility requirement on the general curve.

778 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 6 [7] Lothar Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), no. 3, 523–533, DOI 10.1007/s002200050434. MR1645204 [8] Lothar Göttsche and Vivek Shende, Refined curve counting on complex surfaces, Geom. Topol. 18 (2014), no. 4, 2245– 2307, DOI 10.2140/gt.2014.18.2245. MR3268777 [9] Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461, DOI 10.1007/BF01388741. MR837522 [10] Steven Kleiman and Ragni Piene, Enumerating singu- lar curves on surfaces, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 209–238, DOI 10.1090/conm/241/03637. MR1718146 [11] M. Kontsevich and Yu. Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR1291244 [12] Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368, DOI 10.1007/978-1-4612-4264- 2_12. MR1363062 [13] Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259–367. MR1366548 [14] Francesco Severi, Vorlesungen über algebraische Geometrie: Geometrie auf einer Kurve, Riemannsche Flächen, Abelsche In- tegrale (German), Berechtigte Deutsche Übersetzung von Eugen Löffler. Mit einem Einführungswort von A. Brill. Begleitwort zum Neudruck von Beniamino Segre. Bib- liotheca Mathematica Teubneriana, Band 32, Johnson Reprint Corp., New York-London, 1968. MR0245574 [15] Ravi Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000), no. 1, 53–84, DOI 10.1007/s002291020053. MR1771228

Lucia Caporaso

Credits Opener image is courtesy of Getty. Figures 1–4 are courtesy of Lucia Caporaso. Author photo is courtesy of Dario Birindelli.

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