Geometry of the Severi variety.

*(English)*Zbl 0677.14003The algebraic curves of degree \(d\) in \(\mathbb P^2\) are parametrized by a projective space \(\mathbb P^N\). Denote by \(V=V(d,\delta)\) the locally closed subspace of \(\mathbb P^N\) parametrizing the nodal curves of degree \(d\) and geometric genus \(g\), where each such curve is singular at exactly \(\delta =1/2 (d-1)(d-2)-g\) nodes. The classical Severi problem was to show that \(V\) is irreducible when nonempty; it was solved several years ago by J. Harris [Invent. Math. 84, 445–461 (1986; Zbl 0596.14017)]. Given the irreducibility, we can investigate the structure of suitable full or partial compactifications of \(V\); these are called Severi varieties. The closure in \(\mathbb P^N\), of course, is one, but the boundary of \(V\) in this closure seems forbiddingly complex, because our knowledge of the possible degenerations is quite incomplete. Therefore attention has turned to more accessible possibilities. The object of this fine paper [and its sequel, part II of this paper in Algebraic geometry, Proc. Conf., Sundance Utah 1986, Lect. Notes Math. 1311, 23–50 (1988; Zbl 0677.14004)] is to study, instead, a partial compactification, denoted \(W\), of \(V\), obtained by adjoining the codimension 1 equisingular strata of the closure of \(V\) in \(\mathbb P^N\), and then normalizing. First the authors show that \(W\) is smooth, and then they work out some natural relations in \(\operatorname{Pic}(W)\), using techniques inspired in part by the enumerative geometry of moduli spaces.

Based on the authors’ previous results [cf. “Ideals associated to deformations of singular plane curves”, Trans. Am. Math. Soc. 309, No. 2, 433–468 (1988; Zbl 0707.14022)], the image of \(W\) in \(\mathbb P^N\) is the union of \(V\) and the following codimension 1 strata: CU, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-1\) nodes and one cusp; TN, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-2\) nodes and one tacnode; TR, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-3\) nodes and one ordinary triple point; and \(\Delta\), the locus of reduced curves of geometric genus \(g-1\) with at most two irreducible components and \(\delta+1\) nodes.

Of these four codimension one subschemes of the closure of \(V\), the authors suspected (but did not know then) that the first three are irreducible. After the present article appeared, their irreducibility was established, through a generalization of Harris’ irreducibility theorem, by Ziv Ran [“Families of plane curves and their limits: Enriques’ conjecture and beyond”, Ann. Math. (2) 130, No. 1, 121–157 (1989; Zbl 0704.14018)]. In contrast, \(\Delta\) is reducible, and its components play important parts in the theory.

A complete set of generators for \(\operatorname{Pic}(W)\) is at present unknown, but there are many natural candidates, among them the boundary components mentioned already. The loci of curves with hyperflexes, with flecnodes, with flex bitangents, with nodal tangents tangent to the curve elsewhere, with binodal tangents, with tritangent lines, with three collinear nodes, also come into consideration. Their classes are extrinsic, because they reflect properties of the embeddings of the given curves in the plane. Other potential generators (also extrinsic) depend on the position of given curves relative to points and lines in \(\mathbb P^2\), and so are important in classical enumerative geometry. These include the divisor of curves through a given point, those tangent to a given line, those with a node on a given line, and so on.

We can also consider several intrinsic divisor classes, analogous to the intrinsic classes on the moduli space of stable curves. Over \(W\), there is a universal family \(\mathcal C\) of curves of genus \(g\), and we have two natural divisor classes on \(\mathcal C\): the first Chern class of the relative dualizing sheaf of \(\mathcal C/W\), and the pullback of \(c_1(\mathcal O_{\mathbb P^2}(1))\). We obtain three classes, A, B and C, in \(\operatorname{Pic}(W)\) by taking pairwise products on \(\mathcal C\) and then pushing down.

These intrinsic classes bring some order into the seeming chaos of potential generators, at least over \(\mathbb Q\), because each extrinsic class described above can be expressed (as the authors show) as a rational linear combination of A, B, and C, together with either \(\Delta\) or suitable components of \(\Delta\). As a corollary, the authors show that the original variety \(V\) of node curves is affine. Further, they conjecture that A, B, C, and the components of \(\Delta\) generate \(\operatorname{Pic}(W)\) over \(\mathbb Q\). This conjecture is equivalent to the assertion that \(\operatorname{Pic}(V)\) is torsion.

Partial compactifications of spaces of plane curves are useful for solving enumerative problems [see for example S. L. Kleiman and the reviewer “Enumerative geometry of nodal plane cubics” in Algebraic geometry, Proc. Conf., Sundance/Utah, Lect. Notes Math. 1311, 156–196 (1988; Zbl 0678.14013) and “Enumerative geometry of nonsingular plane cubics”, Algebraic geometry, Proc. Conf. Sundance/Utah 1988, Contemp. Math. 116, 85–113 (1991; Zbl 0753.14045)]. In the latter, we need a different partial compactification, here denoted \(W+\), of the variety \(V\) of nonsingular cubics, in order to obtain proper intersections with the variety of cubics tangent to a given line, because the dual curve also needs to be parametrized by the partial compactification. On \(W^+\), compared to the Severi variety W which it dominates, the locus of triple line degenerations blows up to a divisor. This gives an additional boundary component, crucial for checking the classical enumerative results. Much of the divisor structure of \(W^+\) is reflected in \(W\), however, and this will hold more generally as we pass to plane curves of higher degree.

This article represents a valuable step forward in the study of plane curves, not just for its results, but for the links it provides between the geometry of moduli, the study of the authors’ Severi varieties, and the structure of the somewhat more elaborate parameter spaces needed for an enumerative geometry, envisaged classically but still not carried out, for plane curves of any degree.

Based on the authors’ previous results [cf. “Ideals associated to deformations of singular plane curves”, Trans. Am. Math. Soc. 309, No. 2, 433–468 (1988; Zbl 0707.14022)], the image of \(W\) in \(\mathbb P^N\) is the union of \(V\) and the following codimension 1 strata: CU, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-1\) nodes and one cusp; TN, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-2\) nodes and one tacnode; TR, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-3\) nodes and one ordinary triple point; and \(\Delta\), the locus of reduced curves of geometric genus \(g-1\) with at most two irreducible components and \(\delta+1\) nodes.

Of these four codimension one subschemes of the closure of \(V\), the authors suspected (but did not know then) that the first three are irreducible. After the present article appeared, their irreducibility was established, through a generalization of Harris’ irreducibility theorem, by Ziv Ran [“Families of plane curves and their limits: Enriques’ conjecture and beyond”, Ann. Math. (2) 130, No. 1, 121–157 (1989; Zbl 0704.14018)]. In contrast, \(\Delta\) is reducible, and its components play important parts in the theory.

A complete set of generators for \(\operatorname{Pic}(W)\) is at present unknown, but there are many natural candidates, among them the boundary components mentioned already. The loci of curves with hyperflexes, with flecnodes, with flex bitangents, with nodal tangents tangent to the curve elsewhere, with binodal tangents, with tritangent lines, with three collinear nodes, also come into consideration. Their classes are extrinsic, because they reflect properties of the embeddings of the given curves in the plane. Other potential generators (also extrinsic) depend on the position of given curves relative to points and lines in \(\mathbb P^2\), and so are important in classical enumerative geometry. These include the divisor of curves through a given point, those tangent to a given line, those with a node on a given line, and so on.

We can also consider several intrinsic divisor classes, analogous to the intrinsic classes on the moduli space of stable curves. Over \(W\), there is a universal family \(\mathcal C\) of curves of genus \(g\), and we have two natural divisor classes on \(\mathcal C\): the first Chern class of the relative dualizing sheaf of \(\mathcal C/W\), and the pullback of \(c_1(\mathcal O_{\mathbb P^2}(1))\). We obtain three classes, A, B and C, in \(\operatorname{Pic}(W)\) by taking pairwise products on \(\mathcal C\) and then pushing down.

These intrinsic classes bring some order into the seeming chaos of potential generators, at least over \(\mathbb Q\), because each extrinsic class described above can be expressed (as the authors show) as a rational linear combination of A, B, and C, together with either \(\Delta\) or suitable components of \(\Delta\). As a corollary, the authors show that the original variety \(V\) of node curves is affine. Further, they conjecture that A, B, C, and the components of \(\Delta\) generate \(\operatorname{Pic}(W)\) over \(\mathbb Q\). This conjecture is equivalent to the assertion that \(\operatorname{Pic}(V)\) is torsion.

Partial compactifications of spaces of plane curves are useful for solving enumerative problems [see for example S. L. Kleiman and the reviewer “Enumerative geometry of nodal plane cubics” in Algebraic geometry, Proc. Conf., Sundance/Utah, Lect. Notes Math. 1311, 156–196 (1988; Zbl 0678.14013) and “Enumerative geometry of nonsingular plane cubics”, Algebraic geometry, Proc. Conf. Sundance/Utah 1988, Contemp. Math. 116, 85–113 (1991; Zbl 0753.14045)]. In the latter, we need a different partial compactification, here denoted \(W+\), of the variety \(V\) of nonsingular cubics, in order to obtain proper intersections with the variety of cubics tangent to a given line, because the dual curve also needs to be parametrized by the partial compactification. On \(W^+\), compared to the Severi variety W which it dominates, the locus of triple line degenerations blows up to a divisor. This gives an additional boundary component, crucial for checking the classical enumerative results. Much of the divisor structure of \(W^+\) is reflected in \(W\), however, and this will hold more generally as we pass to plane curves of higher degree.

This article represents a valuable step forward in the study of plane curves, not just for its results, but for the links it provides between the geometry of moduli, the study of the authors’ Severi varieties, and the structure of the somewhat more elaborate parameter spaces needed for an enumerative geometry, envisaged classically but still not carried out, for plane curves of any degree.

Reviewer: Robert Speiser (Provo)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14C22 | Picard groups |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14H99 | Curves in algebraic geometry |

14C20 | Divisors, linear systems, invertible sheaves |

14N05 | Projective techniques in algebraic geometry |

##### Keywords:

enumerative geometry of moduli spaces of curves; generators of Picard group; nodal curves; Severi problem; Severi varieties; divisor classes
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\textit{S. Diaz} and \textit{J. Harris}, Trans. Am. Math. Soc. 309, No. 1, 1--34 (1988; Zbl 0677.14003)

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##### References:

[1] | E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves, Springer-Verlag, New York, 1984. · Zbl 0559.14017 |

[2] | F. Cukierman, Ph.D. thesis, Brown University, 1987. |

[3] | Steven Diaz and Joe Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), no. 2, 433 – 468. · Zbl 0707.14022 |

[4] | David Eisenbud and Joe Harris, When ramification points meet, Invent. Math. 87 (1987), no. 3, 485 – 493. , https://doi.org/10.1007/BF01389239 David Eisenbud and Joe Harris, Existence, decomposition, and limits of certain Weierstrass points, Invent. Math. 87 (1987), no. 3, 495 – 515. · Zbl 0606.14014 · doi:10.1007/BF01389240 · doi.org |

[5] | David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337 – 371. · Zbl 0598.14003 · doi:10.1007/BF01389094 · doi.org |

[6] | G. Ellingrud and S. Stromme, On the homology of the Hilbert scheme of points in the plane, preprint no. 13, Univ. of Oslo, 1984. |

[7] | William Fulton, On nodal curves, Algebraic geometry — open problems (Ravello, 1982) Lecture Notes in Math., vol. 997, Springer, Berlin, 1983, pp. 146 – 155. · Zbl 0514.14012 · doi:10.1007/BFb0061642 · doi.org |

[8] | William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005 |

[9] | Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445 – 461. · Zbl 0596.14017 · doi:10.1007/BF01388741 · doi.org |

[10] | Joe Harris, Curves and their moduli, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 99 – 143. |

[11] | Z. Ran, Degenerations of linear systems, preprint. |

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