Abstract
Given a smooth k-variety Y (where k is a field of arbitrary characteristic) and a linear systemL on Y we study the dimension of the singular locus of the general element ofL, both inside and outside the base locus B ofL. We interpret these results from the point of view of the transversality theory, and we improve a result by Speiser about the not too ramified morphisms. Moreover, we show that our results can be applied in some cases where a criterion by Zhang, for the smoothness of the general element ofL, fails.
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References
C. Cumino—S. Greco—M. Manaresi,An axiomatic approach to the second theorem of Bertini, Jour. of Algebra,98 (1986), pp. 171–182.
S. Greco—P. Valabrega,On the Singular Locus of a General Complete Intersection through a Variety in Projective Space, Boll. UMI Alg. e Geom. Serie VI, Vol. II,1 (1983), pp. 113–145.
R. Hartshorne,Algebraic Geometry, GTM52, Springer-Verlag, New York-Heidelberg-Berlin (1977).
S. L. Kleiman,Transversality of the general translate, Compos. Math.,28 (1973), pp. 287–297.
D. Laksov—R. Speiser,Transversality Criteria in any Characteristic, LNM1436, Proc. 1987 Sitiges conference on enumerative geometry, pp. 139–150 Springer-Verlag, New York-Heidelberg-Berlin (1990).
D. Laksov—R. Speiser,Determinantal Criteria for Transversality of Morphism, Pacific Jour. of Math., Vol.156, N. 2 (1992), pp. 307–328.
H. Matsumura,Commutative Ring Theory, Cambridge studies in adv. math.,8, Cambridge University Press, Cambridge-New York-Melbourne (1930).
R. Speiser,Transversality Theorems for families of maps, LNM1311 Proc. 1986 Sundance Conference, pp. 235–252 Springer-Verlag, New York-Heidelberg-Berlin (1988).
M. L. Spreafico,Axiomatic Theory for Transversality and Bertini type Theorems, Arch. der Math.,70 (1998), pp. 407–424.
M. L. Spreafico,Bertini Type Theorems for vector bundles in any characteristics, Comm. in Alg.,24 (1996), pp. 4147–4157.
B. Zhang,Théorèmes du type Bertini en caractéristique positive, Arch. Math.,64 (1995), pp. 209–215.
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Entrata in Redazione il 15 settembre 1997, e in versione definitiva il 28 ottobre 1999.
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Spreafico, M.L. Linear systems, singularity and not too ramified morphisms. Annali di Matematica pura ed applicata 179, 295–307 (2001). https://doi.org/10.1007/BF02505960
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DOI: https://doi.org/10.1007/BF02505960