On parameterizations of plane rational curves and their syzygies. (English) Zbl 1342.14064
The paper under review describes a relationship between the geometry of a rational plane curve \(C\subset\mathbb P^2\) of degree \(d\) and its splitting type. Such a curve \(C\) is the image of a generically injective map \(f:\mathbb P^1\to\mathbb P^2\) given by polynomials \(f_0, f_1, f_2 \in S_d\), where \(S\) is the homogeneous coordinate ring for \(\mathbb P^1\). Setting \(I =(f_0,f_1,f_2)\subset S\), the kernel of the natural map \(S(-d)^3\to I\) has the form \(S(-k-d)\oplus S(k-2d)\), which defines the splitting type \((k,d-k)\). For \(C\) general, the splitting type is \((\lfloor d/2 \rfloor, \lceil d/2 \rceil)\). M. G. Ascenzi showed that if \(C\) has a singularity of multiplicity \(m \geq (d-1)/2\), then the splitting type is \((m,d-m)\) [Commun. Algebra 16, No. 11, 2193–2208 (1988; Zbl 0675.14010)]: moreover if \(C\) has splitting type \((1,d-1)\), then \(C\) must have a point of multiplicity \(d-1\); if \(C\) has splitting type \((2,d-2)\), then either \(C\) has a point of multiplicity \(d-2\) or \(C\) is a nodal projection of a smooth curve lying on a quadric surface in \(\mathbb P^3\).
The authors extend these classifications by proving their main theorem: A rational curve \(C \subset \mathbb P^2\) has splitting type \((k,d-k)\) with \(1 \leq k \leq d/2\) if and only if \(C\) is the projection of a rational curve \(D \subset \mathbb P^{k+1}\) lying on a rational normal scroll \(S\) from \(k-1\) points outside of \(S\). They first construct \(D^\prime \subset \mathbb P^2 \times \mathbb P^1\) as the graph of the map \(f\) and use the lowest degree form in one of the bi-graded pieces of the ideal to construct \(S^\prime\), using partial derivatives to show that both are smooth. Then they interpret \(S^\prime\) as a Hirzebruch surface and map it into \(\mathbb P^{k+1}\) using a suitable linear system and let \(D \subset S\) be the corresponding images. Their method shows that if \(S\) is smooth, then \(C\) has no point of multiplicity \(m \geq (d-1)/2\); otherwise \(S\) is a cone and \(D\) passes through the vertex with multiplicity \(d-k\), when \(C\) has a point of multiplicity \(d-k\).
The authors extend these classifications by proving their main theorem: A rational curve \(C \subset \mathbb P^2\) has splitting type \((k,d-k)\) with \(1 \leq k \leq d/2\) if and only if \(C\) is the projection of a rational curve \(D \subset \mathbb P^{k+1}\) lying on a rational normal scroll \(S\) from \(k-1\) points outside of \(S\). They first construct \(D^\prime \subset \mathbb P^2 \times \mathbb P^1\) as the graph of the map \(f\) and use the lowest degree form in one of the bi-graded pieces of the ideal to construct \(S^\prime\), using partial derivatives to show that both are smooth. Then they interpret \(S^\prime\) as a Hirzebruch surface and map it into \(\mathbb P^{k+1}\) using a suitable linear system and let \(D \subset S\) be the corresponding images. Their method shows that if \(S\) is smooth, then \(C\) has no point of multiplicity \(m \geq (d-1)/2\); otherwise \(S\) is a cone and \(D\) passes through the vertex with multiplicity \(d-k\), when \(C\) has a point of multiplicity \(d-k\).
Reviewer: Scott Nollet (Fort Worth)
MSC:
| 14H45 | Special algebraic curves and curves of low genus |
| 14H50 | Plane and space curves |
| 14N05 | Projective techniques in algebraic geometry |
Citations:
Zbl 0675.14010Software:
CoCoAReferences:
| [1] | J.Abbott, A. M.Bigatti, and G.Lagorio, CoCoA‐5: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it, 2008. |
| [2] | M. G.Ascenzi, The restricted tangent bundle of a rational curve on a quadric in \(\mathbb{P}^3\), Proc. Amer. Math. Soc.98(4), 561-566 (1986). · Zbl 0616.14027 |
| [3] | M. G.Ascenzi, The restricted tangent bundle of a rational curve in \(\mathbb{P}^2\), Comm. Algebra16(11), 2193-2208 (1988). · Zbl 0675.14010 |
| [4] | E.Ballico and Ph.Ellia, The maximal rank conjecture for non special curves in \(\mathbb{P}^n\), Math. Z.196, 355-367 (1987). · Zbl 0631.14029 |
| [5] | A.Bernardi, Apolar ideal and normal bundle of rational curves, http://arxiv.org/abs/1203.4972, 03 2012. |
| [6] | A.Bernardi, Normal bundle of rational curves and Waring decomposition, J. Algebra400, 123-141 (2014). · Zbl 1318.14030 |
| [7] | G.Birkhoff, A theorem on matrices of analytic functions, Math. Ann.74(1), 122-133 (1913). · JFM 44.0469.02 |
| [8] | I.Coskun, Degenerations of surface scrolls and the gromov‐witten invariants of grassmannians, J. Algebraic Geom.15, 223-284 (2006). · Zbl 1105.14072 |
| [9] | D.Cox, A. R.Kustin, C.Polini, and B.Ulrich, A Study of Singularities on Rational Curves via Syzygies, Memoirs of the American Mathematical Society Vol. 1045 (2013). · Zbl 1305.14014 |
| [10] | D.Cox, T. W.Sederburg, and F.Chen, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Des.15, 803-827 (1998). · Zbl 0908.68174 |
| [11] | D. A.Cox and A. A.Iarrobino, Strata of rational space curves, Comput. Aided Geom. Design32(C), 50-68 January (2015). · Zbl 1417.14009 |
| [12] | D.Eisenbud and A.Van de Ven, On the normal bundles of smooth rational space curves, Math. Ann.256(4), 453-463 (1981). · Zbl 0443.14015 |
| [13] | D.Eisenbud and A.Van de Ven, On the variety of smooth rational space curves with given degree and normal bundle, Invent. Math.67(1), 89-100 (1982). · Zbl 0492.14016 |
| [14] | D.Eisenbud and J.Harris, On varieties of minimal degree (a centennial account), In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., Vol. 46 (Amer. Math. Soc., Providence, RI, 1987), pp. 3-13. · Zbl 0646.14036 |
| [15] | F.Ghione and G.Sacchiero, Normal bundle of rational curves in \(\mathbb{P}^3\), Manuscripta Math.33(2), 111-128 (1980/81). · Zbl 0496.14021 |
| [16] | AGimigliano, B.Harbourne, and M.Idà, Betti numbers for fat point ideals in the plane: a geometric approach, Trans. Amer. Math. Soc.361, 1103-1127 (2009). · Zbl 1170.14006 |
| [17] | AGimigliano, B.Harbourne, and M.Idà, The role of the cotangent bundle in resolving ideals of fat points in the plane, J. Pure Appl. Algebra213, 203-214 (2009). · Zbl 1161.14012 |
| [18] | AGimigliano, B.Harbourne, and M.Idà, Stable postulation and stable ideal generation: conjectures for fat points in the plane, Bull. Belg. Math. Soc. Simon Stevin16(5), 853-860 (2009). · Zbl 1187.14010 |
| [19] | AGimigliano, B.Harbourne, and M.Idà, On plane rational curves and the splitting of the tangent bundle, Ann. Sc. Norm. Super. Pisa Vl. Sci.XII(5), 1-35 (2013). |
| [20] | A.Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math.79(1), 121-138 (1957). · Zbl 0079.17001 |
| [21] | R.Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, (Springer‐Verlag, New York‐Heidelberg, 1977). · Zbl 0367.14001 |
| [22] | A.Hirschowitz, Sur la postulation générique des courbes rationnelles, Acta. Math.146, 209-230 (1981). · Zbl 0475.14027 |
| [23] | K.Hulek, The normal bundle of a curve on a quadric, Math. Ann.258, 201-206 (1981). · Zbl 0458.14011 |
| [24] | A. A.Iarrobino, Strata of vector spaces of forms in R = K[x,y], and of rational curves in , Bull. Braz. Math. Soc. (N.S.)45(4), 711-725 (2014). · Zbl 1317.14026 |
| [25] | L.Ramella, La stratification du schéma de Hilbert des courbes rationnelles de \(\mathbf{P}^n\) par le fibré tangent restreint, C. R. Acad. Sci. Paris Sér. I Math.311(3), 181-184 (1990). · Zbl 0721.14014 |
| [26] | C.Segre, Sulle Rigate Razionali in Uno Spazio Lineare Qualunque (Loescher, 1884). · JFM 16.0604.01 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
