Blowup algebras of rational normal scrolls

Sammartano, Alessio

doi:10.1090/tran/7689

Blowup algebras of rational normal scrolls
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by Alessio Sammartano PDF

Trans. Amer. Math. Soc. 373 (2020), 797-818 Request permission

Abstract:

We determine the equations of the blowup of $\mathbb {P}^{n}$ along a $d$-fold rational normal scroll $\mathcal {S}$, and we prove that the Rees ring and special fiber ring of $\mathcal {S}\subseteq \mathbb {P}^{n}$ are Koszul algebras.

References

Grigoriy Blekherman, Gregory G. Smith, and Mauricio Velasco, Sums of squares and varieties of minimal degree, J. Amer. Math. Soc. 29 (2016), no. 3, 893–913. MR 3486176, DOI 10.1090/jams/847
Stefan Blum, Subalgebras of bigraded Koszul algebras, J. Algebra 242 (2001), no. 2, 795–809. MR 1848973, DOI 10.1006/jabr.2001.8804
Winfried Bruns, Aldo Conca, and Matteo Varbaro, Relations between the minors of a generic matrix, Adv. Math. 244 (2013), 171–206. MR 3077870, DOI 10.1016/j.aim.2013.05.004
Winfried Bruns, Aldo Conca, and Matteo Varbaro, Maximal minors and linear powers, J. Reine Angew. Math. 702 (2015), 41–53. MR 3341465, DOI 10.1515/crelle-2013-0026
Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
Laurent Busé, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra 321 (2009), no. 8, 2317–2344. MR 2501523, DOI 10.1016/j.jalgebra.2009.01.030
Michael L. Catalano-Johnson, The possible dimensions of the higher secant varieties, Amer. J. Math. 118 (1996), no. 2, 355–361. MR 1385282, DOI 10.1353/ajm.1996.0012
Marc Chardin, Powers of ideals and the cohomology of stalks and fibers of morphisms, Algebra Number Theory 7 (2013), no. 1, 1–18. MR 3037888, DOI 10.2140/ant.2013.7.1
A. Conca, Koszul algebras and Gröbner bases of quadrics, Proceedings of the 29th Symposium on Commutative Algebra in Nagoya, Japan (2007), pp. 127–133.
Aldo Conca, Jürgen Herzog, and Giuseppe Valla, Sagbi bases with applications to blow-up algebras, J. Reine Angew. Math. 474 (1996), 113–138. MR 1390693, DOI 10.1515/crll.1996.474.113
David A. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), no. 1-3, 23–36. MR 2394983, DOI 10.1016/j.tcs.2007.10.012
S. Dale Cutkosky, Jürgen Herzog, and Ngô Viêt Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), no. 3, 243–261. MR 1711319, DOI 10.1023/A:1001559912258
A. V. Doria, S. H. Hassanzadeh, and A. Simis, A characteristic-free criterion of birationality, Adv. Math. 230 (2012), no. 1, 390–413. MR 2900548, DOI 10.1016/j.aim.2011.12.005
David Eisenbud and Joe Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR 927946, DOI 10.1090/pspum/046.1/927946
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
Olga Holtz and Bernd Sturmfels, Hyperdeterminantal relations among symmetric principal minors, J. Algebra 316 (2007), no. 2, 634–648. MR 2358606, DOI 10.1016/j.jalgebra.2007.01.039
Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, Rational normal scrolls and the defining equations of Rees algebras, J. Reine Angew. Math. 650 (2011), 23–65. MR 2770555, DOI 10.1515/CRELLE.2011.002
Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, Blowups and fibers of morphisms, Nagoya Math. J. 224 (2016), no. 1, 168–201. MR 3572752, DOI 10.1017/nmj.2016.34
Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, The equations defining blowup algebras of height three Gorenstein ideals, Algebra Number Theory 11 (2017), no. 7, 1489–1525. MR 3697146, DOI 10.2140/ant.2017.11.1489
Susan Morey and Bernd Ulrich, Rees algebras of ideals with low codimension, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3653–3661. MR 1343713, DOI 10.1090/S0002-9939-96-03470-3
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Luke Oeding, Set-theoretic defining equations of the variety of principal minors of symmetric matrices, Algebra Number Theory 5 (2011), no. 1, 75–109. MR 2833786, DOI 10.2140/ant.2011.5.75
T. Kyle Petersen, Pavlo Pylyavskyy, and David E. Speyer, A non-crossing standard monomial theory, J. Algebra 324 (2010), no. 5, 951–969. MR 2659207, DOI 10.1016/j.jalgebra.2010.05.001
Shlomi Reuveni, Catalan’s trapezoids, Probab. Engrg. Inform. Sci. 28 (2014), no. 3, 353–361. MR 3214794, DOI 10.1017/S0269964814000047
Tim Römer, Homological properties of bigraded algebras, Illinois J. Math. 45 (2001), no. 4, 1361–1376. MR 1895463
Francesco Russo, On the geometry of some special projective varieties, Lecture Notes of the Unione Matematica Italiana, vol. 18, Springer, Cham; Unione Matematica Italiana, Bologna, 2016. MR 3445582, DOI 10.1007/978-3-319-26765-4
Francesco Russo and Aron Simis, On birational maps and Jacobian matrices, Compositio Math. 126 (2001), no. 3, 335–358. MR 1834742, DOI 10.1023/A:1017572213947
Francisco Santos, Christian Stump, and Volkmar Welker, Noncrossing sets and a Grassmann associahedron, Forum Math. Sigma 5 (2017), Paper No. e5, 49. MR 3610869, DOI 10.1017/fms.2017.1
Aron Simis, Cremona transformations and some related algebras, J. Algebra 280 (2004), no. 1, 162–179. MR 2081926, DOI 10.1016/j.jalgebra.2004.03.025
Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
Bernd Sturmfels and Neil White, Gröbner bases and invariant theory, Adv. Math. 76 (1989), no. 2, 245–259. MR 1013672, DOI 10.1016/0001-8708(89)90053-4