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Enumerative Geometry of Plane Curves

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Enumerative Geometry of Plane Curves 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 algebraic geometry is the 2 complex projective space, ℙ푟, viewed as a topological space and, by definition, two nonzero polynomials determine with the Zariski topology. 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.
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