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Geometric properties of projective manifolds of small degree

Published online by Cambridge University Press:  02 December 2015

SIJONG KWAK  and
JINHYUNG PARK
SIJONG KWAK
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon, Korea. e-mail: sjkwak@kaist.ac.kr
JINHYUNG PARK
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon, Korea. Current address: School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea. e-mail: parkjh13@kias.re.kr
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Abstract

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb{P}$r of dimension n and degree d ⩽ n(r − n) + 2 is Calabi–Yau, and give an example that shows this bound is also sharp.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 2 , March 2016 , pp. 257 - 277
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[AHL]Artebani, M., Hausen, J. and Laface, A.On Cox rings of K3 surfaces. Composition Math. 146 (2010), 964–998.CrossRefGoogle Scholar
[BS]Beltrametti, M. C. and Sommese, A. J.The adjunction theory of complex projective varieties, de Gruyter Expositions in Mathematics 16Walter de Gruyter and Co., Berlin, (1995).CrossRefGoogle Scholar
[BCHM]Birkar, C., Cascini, P., Hacon, C. and McKernan, J.Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), 405–468.CrossRefGoogle Scholar
[Bui]Buium, A.On surfaces of degree at most 2n + 1 in ${\mathbb{P}}$n in Algebraic Geometry (Bucharest 1982), Lecture Notes in Math. 1056 (Springer-Verlag, Berlin, 1984), 47–67.Google Scholar
[But]Butler, D. C.Normal generation of vector bundles over a curve. J. Differential Geom. 39 (1994), 1–34.CrossRefGoogle Scholar
[E]Ein, L.Nondegenerate surfaces of degree n + 3 in ${\mathbb{P}}_{\mathbb{C}}^n$. J. Reine Angew. Math. 351 (1984), 1–11.Google Scholar
[EH]Eisenbud, D. and Harris, J.On varieties of minimal degree (a centennial account) In Algebraic geometry, Bowdoin (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46 (1987), Part 1 (Amer. Math. Soc., Providence, RI), 3–13.Google Scholar
[FL]Fania, M. L. and Livorni, E. L.Degree nine manifolds of dimension greater than or equal to 3. Math. Nachr. 169 (1994), 117–134.CrossRefGoogle Scholar
[F1]Fujita, T.On the structure of polarized manifolds with total deficiency one I. J. Math. Soc. Japan 32 (1980), 709–725.CrossRefGoogle Scholar
[F2]Fujita, T.On the structure of polarized manifolds with total deficiency one II. J. Math. Soc. Japan 33 (1981), 415–434.CrossRefGoogle Scholar
[Fu]Fukuma, Y.On the sectional geometric genus of quasi-polarized varieties. II. Manuscripta Math. 113 (2004), 211–237.CrossRefGoogle Scholar
[GHS]Graber, T., Harris, J. and Starr, J.Families of rationally connected varieties. J. Amer. Math. Soc. 16 (2003), 57–67.CrossRefGoogle Scholar
[HM]Hacon, C. and McKernan, J.On Shokurov's rational connectedness conjecture. Duke. Math. J. 138 (2007), 119–136.CrossRefGoogle Scholar
[Hr]Harris, J.A bound on the geometric genus of projective varieties. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 35–68.Google Scholar
[Ht]Hartshorne, R.Varieties of small codimension in projective space. Bull. Amer. Math. Soc. 80 (1974), 1017–1032.CrossRefGoogle Scholar
[HK]Hu, Y. and Keel, S.Mori dream spaces and GIT. Michigan Math. J. 48 (2000), 331–348.CrossRefGoogle Scholar
[II]Ilic, B.Geometric properties of the double-point divisor. Trans. Amer. Math. Soc. 350 (1998), 1643–1661.CrossRefGoogle Scholar
[Io1]Ionescu, P.Embedded projective varieties of small invariants. In Algebraic Geometry (Bucharest 1982), Lecture Notes in Math. 1056 (Springer-Verlag, Berlin, 1984), 142–186.Google Scholar
[Io2]Ionescu, P.On varieties whose degree is small with respect to codimension. Math. Ann. 271 (1985), 339–348.CrossRefGoogle Scholar
[Io3]Ionescu, P.Embedded projective varieties of small invariants III. In Algebraic Geometry (L'Aquila 1988), Lecture Notes in Math. 1417 (Springer-Verlag, Berlin, 1990), 138–154.CrossRefGoogle Scholar
[Io4]Ionescu, P.On manifolds of small degree. Comment. Math. Helv. 83 (2008), 927–940.CrossRefGoogle Scholar
[IT1]Ionescu, P. and Toma, M.Boundedness for some special families of embedded manifolds. Contemp. Math. 162 (1994), 215–225.CrossRefGoogle Scholar
[IT2]Ionescu, P. and Toma, M.On very ample vector bundles on curves. Internat. J. Math. 8 (1997), 633–643.CrossRefGoogle Scholar
[IP]Iskovskikh, V. A. and Prokhorov, Y. G.Algebraic geometry V: Fano varieties. Encyclopaedia Math. Sci. 47 (Springer, Berlin, 1999).Google Scholar
[It]Ito, A.Examples of Mori dream spaces with Picard number two. Manuscripta Math. 145 (2014), 243–254.CrossRefGoogle Scholar
[KM]Kollár, J. and Mori, S.Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134 (Cambridge University Press, Cambridge 1998).Google Scholar
[LN]Lanteri, A. and Novelli, C.Ample vector bundles of small Δ-genera. J. Algebra 323 (2010), 671–697.CrossRefGoogle Scholar
[L1]Lazarsfeld, R.Positivity in algebraic geometry I. Classical Setting: line bundles and linear series. A Series of Modern Surveys in Math. 48 (Springer-Verlag, Berlin, 2004).Google Scholar
[L2]Lazarsfeld, R.Positivity in algebraic geometry II. Positivity for vector bundles, and multiplier ideals, A Series of Modern Surveys in Math. 49 (Springer-Verlag, Berlin, 2004).CrossRefGoogle Scholar
[Me]Mella, M.#-Minimal models of uniruled 3-folds. Math. Z. 242 (2002), 687–707.CrossRefGoogle Scholar
[MM]Mori, S. and Mukai, S.Classification of Fano 3-folds with B 2 ⩾ 2. Manuscripta Math. 36 (1981), 147–162. See also Erratum: “Classification of Fano 3-folds with B 2 ⩾ 2”. Manuscripta Math. 110 (2003), 407.CrossRefGoogle Scholar
[Mu]Mukai, S.Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 3000–3002.CrossRefGoogle ScholarPubMed
[N]Noma, A.Generic inner projections of projective varieties and an application to the positivity of double point divisors. Trans. Amer. Math. Soc. 336 (2014), 4603–4623.CrossRefGoogle Scholar
[Og]Oguiso, K.Quartic K3 surfaces and Cremona transformations. In Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds (Laza, R.et al. (eds.)). Fields Institue Communications 67 (Springer-Verlag, Berlin, 2013).Google Scholar
[Ot]Ottaviani, G.On 3-folds in ${\mathbb{P}}$5 which are scrolls. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 451–471.Google Scholar
[TVAV]Testa, D., Várilly-Alvarado, A. and Velasco, M.Big rational surfaces. Math. Ann. 351 (2011), 95–107.CrossRefGoogle Scholar
[W1]Wiśniewski, J.On a conjecture of Mukai. Manuscripta Math. 68 (1990), 135–141.CrossRefGoogle Scholar
[W2]Wiśniewski, J.Fano 4-folds of index 2 with b 2 ⩾ 2. A contribution to Mukai classification. Bull. Polish Acad. Sci. Math. 38 (1990), 173–184.Google Scholar
[Z]Zak, F. L.Castelnuovo bounds for higher-dimensional varieties. Composoition Math. 148 (2012), 1085–1132.CrossRefGoogle Scholar