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Even sets of nodes on sextic surfaces. (English) Zbl 1143.14030

Let \(F\) be a nodal complex surface in \(\mathbb P^3\) of degree \(d\), i.e., \(F\) has only nodes (ordinary double points) as singularities. A classical problem, still open for \(d>6\), is to find the maximum number of nodes \(\mu(d)\) that such a surface \(F\) can have. In order to show that \(\mu(5)\) is achieved by Togliatti quintic and that it is \(31\), A. Beauville [Algebraic Geometry (Angers, 1979), Sijthoff and Noordhoff, 207–215 (1980; Zbl 0445.14016)], introduced the notion of an even set of nodes. This notion can be derived from the coding theory framework, which was later used by D. B. Jaffe and D. Ruberman [J. Algebraic Geom. 6, 151–168 (1997; Zbl 0884.14015)] to show that \(\mu(6)\) is achieved by Barth sextic and that it is \(65\).
The main results of the paper under review are the proofs of two theorems: A and B. Theorem A states: Let \(F\) be a nodal surface of degree \(d=6\) in \(\mathbb P^3\) with an even set of \(t\) nodes. Then \(t \in \{24,32,40,56\}\). These four possibilities occur and can be explicitly described.
Partial results were known, e.g. an even set of \(48\) nodes was ruled out by Jaffe and Ruberman in the previously-quoted paper. Considering \(d<6\) as well, cf. e.g. G. Casnati and F. Catanese [J. Differ. Geom. 47, 237–256 (1997; Zbl 0896.14017)], the situation for \(d \leq 6\) is as follows: \[ d=3, \;t=4; \quad d=4, \;t \in \{8,16\}; \quad d=5, \;t \in \{16,20\};\quad d=6, \;t \in \{24,32,40,56\}. \] Theorem B concerns results obtained for a family of nodal sextic surfaces with \(56\) nodes forming an even set.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14N10 Enumerative problems (combinatorial problems) in algebraic geometry

Software:

Macaulay2
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References:

[1] Barth, W.: Two projective surfaces with many nodes, admitting the symmetries of the icosahedron. J. Algebraic Geom. 5 , 173-186 (1996) · Zbl 0860.14032
[2] Barth, W.: Counting singularities of quadratic forms on vector bundles. In: Vector Bun- dles and Differential Equations (Nice, 1979), Progr. Math. 7, Birkhäuser, Boston, MA, 1-19 (1980) · Zbl 0442.14021
[3] Basset, A. B.: The maximum number of double points on a surface. Nature 73 , 246 (1905) · JFM 37.0646.03
[4] Beauville, A.: Sur le nombre maximum de points doubles d’une surface dans 3 P (\mu (5) = 31). In: Algebraic Geometry (Angers, 1979), A. Beauville (ed.), Sijthoff &amp; Noordhoff, 207-215 (1980) · Zbl 0445.14016
[5] Beilinson, A.: Coherent sheaves on N P and problems of linear algebra. Functional Anal. Appl. 12 , 214-216 (1978) · Zbl 0424.14003 · doi:10.1007/BF01681436
[6] Burns, D. M. Jr., Wahl, J. M.: Local contributions to global deformations of surfaces. Invent. Math. 26 , 67-88 (1974) · Zbl 0288.14010 · doi:10.1007/BF01406846
[7] Casnati, G., Catanese, F.: Even sets of nodes are bundle symmetric. J. Differen- tial Geom. 47 , 237-256 (1997). Erratum, J. Differential Geom. 50 , 415 (1998) · Zbl 0896.14017
[8] Catanese, F.: Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications. Invent. Math. 63 , 433-465 (1981) · Zbl 0472.14024 · doi:10.1007/BF01389064
[9] Catanese, F.: Generalized Kummer surfaces and differentiable invariants of Noether- Horikawa surfaces. I. In: Manifolds and Geometry (Pisa, 1993), Sympos. Math. 36, Cambridge Univ. Press, Cambridge, 132-177 (1996) · Zbl 0872.14031
[10] Catanese, F., Ceresa, G.: Constructing sextic surfaces with a given number d of nodes. J. Pure Appl. Algebra 23 , 1-12 (1982) · Zbl 0484.14011 · doi:10.1016/0022-4049(82)90073-1
[11] Cayley, A.: A memoir on cubic surfaces. Trans. London Math. Soc. 159 , 231-326 (1869) · JFM 02.0576.01
[12] Cayley, A.: A third memoir on quartic surfaces. Proc. London Math. Soc. 3 , 234-266 (1871) · JFM 03.0391.02
[13] Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Grad. Texts in Math. 150, Springer, New York (1995) · Zbl 0819.13001
[14] Endraß, S.: Minimal even sets of nodes. J. Reine Angew. Math. 503 , 87-108 (1998) · Zbl 0911.14017 · doi:10.1515/crll.1998.101
[15] Grayson, D., Stillman, M.: Macaulay 2-a software system for algebraic geometry and commutative algebra. http://www.math.uiuc.edu/Macaulay2 (1999)
[16] Jaffe, D. B., Ruberman, D.: A sextic surface cannot have 66 nodes. J. Algebraic Geom. 6 , 151-168 (1997) · Zbl 0884.14015
[17] Kobayashi, S.: On moduli of vector bundles. In: Complex Geometry and Analysis (Pisa, 1988), Lecture Notes in Math. 1422, Springer, Berlin, 45-57 (1990) · Zbl 0718.32020
[18] Kummer, E.: Ueber diejenigen Flächen, welche mit ihren reciprok polaren Flächen von gleicher Ordnung sind und dieselben Singularitäten besitzen. Berl. Monatsber. 1878 , 25-36 · JFM 10.0545.01
[19] MacWilliams, F. J., Sloane, N. J. A.: The Theory of Error-Correcting Codes I, II. North-Holland Math. Library 16, North-Holland, Amsterdam (1977) · Zbl 0369.94008
[20] Miyaoka, Y.: The maximal number of quotient singularities on surfaces with given nu- merical invariants. Math. Ann. 268 , 159-171 (1984) · Zbl 0521.14013 · doi:10.1007/BF01456083
[21] Schreyer, F.-O.: Small fields in constructive algebraic geometry. In: Moduli of Vector Bundles (Sanda and Kyoto, 1994), Lecture Notes in Pure Appl. Math. 179, Dekker, 221-228 (1996) · Zbl 0876.14040
[22] Schreyer, F.-O., Tonoli, F.: Needles in a haystack: special varieties via small fields. In: Computations in Algebraic Geometry with Macaulay 2, D. Eisenbud et al. (eds.), Springer, 251-279 (2002). · Zbl 0994.14037
[23] Tjurina, G. N.: Resolution of singularities of plane (= flat) deformations of double rational points. Funktsional. Anal. i Prilozhen. 4 , no. 1, 77-83 (1970) (in Russian) · Zbl 0221.32008 · doi:10.1007/BF01075621
[24] Togliatti, E. G.: Sulle forme cubiche dello spazio a cinque dimensioni aventi il massimo numero finito di punti doppi. In: Scritti Mat. off. a Luigi Berzolari, 577-593 (1936) · Zbl 0016.22102
[25] Togliatti, E. G.: Una notevole superficie di 5o ordine con soli punti doppi iso- lati. Vierteljschr. Naturforsch. Ges. Zürich 85 , 127-132 (1940) · Zbl 0023.36601
[26] Wahl, J.: Nodes on sextic hypersurfaces in P 3. J. Differential Geom. 48 , 439-444 (1998) · Zbl 0931.14025
[27] Walter, C.: Pfaffian subschemes. J. Algebraic Geom. 5 (1996), 671-704. · Zbl 0864.14032
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