Curves In P3 Research Articles

On the weak Lefschetz property for height four equigenerated complete intersections

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P 3 \mathbb P^3 , and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3 , 4 d=3,4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P 3 \mathbb P^3 .

Irreducible contact curves via graph stratification

We prove that the moduli space of contact stable maps to P2n+1 of degree d admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in P2n+1 with any Schubert condition. We give explicitely some of these invariants for P3 and P5. We give another proof of the formula for the number of plane contact curves in P3 meeting the appropriate number of lines.

Space curves on surfaces with ordinary singularities

We show that smooth curves in the same biliaison class on a hypersurface in $\mathbf{P}^3$ with ordinary singularities are linearly equivalent. We compute the invariants $h^0(\mathscr{I}_C(d))$, $h^1(\mathscr{I}_C(d))$ and $h^1(\mathscr{O}_C(d))$ of a curve $C$ on such a surface $X$ in terms of the cohomologies of divisors on the normalization of $X$. We then study general projections in $\mathbf{P}^3$ of curves lying on the rational normal scroll $S(a,b)\subset\mathbf{P}^{a+b+1}$. If we vary the curves in a linear system on $S(a,b)$ as well as the projections, we obtain a family of curves in $\mathbf{P}^3$. We compute the dimension of the space of deformations of these curves in $\mathbf{P}^3$ as well as the dimension of the family. We show that the difference is a linear function in $a$ and $b$ which does not depend on the linear system. Finally, we classify maximal rank curves on ruled cubic surfaces in $\mathbf{P}^3$. We prove that the general projections of all but finitely many classes of projectively normal curves on $S(1,2)\subset\mathbf{P}^4$ fail to have maximal rank in $\mathbf{P}^3$. These give infinitely many classes of counter-examples to a question of Hartshorne.

Counting planar curves in [formula omitted] with degenerate singularities

In this paper, we consider the following question: how many degree d curves are there in P3 (passing through the right number of generic lines and points), whose image lies inside a P2, having δ nodes and one singularity of codimension k? We obtain an explicit formula for this number when δ+k≤4 (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in the paper by A. Zinger [30] and which is further pursued by S. Basu and the second author in [1], [2] and [3]. Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by S. Kleiman and R. Piene (in [17]) and by T. Laarakker (in [19]). We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author (in [22]), where they compute the characteristic number of rational planar curves in P3 with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande in [23], regarding the enumerativity of BPS numbers for P3.

Rational cuspidal curves in a moving family of ℙ2

Abstract In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).

Tritangents and their space sextics

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve in P3 and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up P2 at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in P2 from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi [8] which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma.

Families of Elliptic Curves in P3 and Bridgeland Stability

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in the three-dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blowup with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.

Enumeration of rational curves in a moving family of [formula omitted

We obtain a recursive formula for the number of rational degree d curves in P3, whose image lies in a P2, intersecting r lines and s points, where r+2s=3d+2. This can be viewed as a family version of the classical question of counting rational curves in P2. We verify that our numbers are consistent with those obtained by T. Laarakker, where he studies the parallel question of counting δ-nodal degree d curves in P3 whose image lies inside a P2. Our numbers give evidence to support the conjecture, that the polynomials obtained by T. Laarakker are enumerative when d≥1+[δ2], which is analogous to the Göttsche threshold for counting nodal curves in P2.

Volumes de Fano faibles obtenus par éclatement d’une courbe de P3

We characterize smooth curves in P3 whose blowup produces a threefold with big and nef anticanonical divisor.

Stable Ulrich bundles on Fano threefolds with Picard number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of P3, Q (smooth quadric in P4), V3 (smooth cubic in P4) or V4 (complete intersection of two quadrics in P5) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in P3. Also, we prove that there exist stable rank two Ulrich bundles with c1=3H on a generic member of this deformation class.

On the intersection of ACM curves in [formula omitted

Bezout's theorem gives us the degree of intersection of two properly intersecting projective varieties. As two curves in P3 never intersect properly, Bezout's theorem cannot be directly used to bound the number of intersection points of such curves. In this work, we bound the maximum number of intersection points of two integral ACM curves in P3. The bound that we give is in many cases optimal as a function of only the degrees and the initial degrees of the curves.

Borel degenerations of arithmetically Cohen–Macaulay curves in ℙ3

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen–Macaulay (ACM) codimension two schemes in ℙn. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in ℙ3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

On set-theoretic complete intersection monomial curves in ℙ3

In this paper, we prove that there are infinitely many set-theoretic complete intersection monomial curves $\bar C$ (l p , l q , ap + bq) in the projective space ℙ3 for every suitably chosen integers l, a, b. We also give explicitly the polynomials defining the corresponding surfaces.

Enumeration of surfaces containing a curve of low degree

Noether–Lefschetz theory tells us that a very general surface of degree at least 4 in P3 has Picard group Z, i.e.,it contains only curves which are complete intersections with other surfaces. Let W be some irreducible subvariety of the Hilbert scheme of curves in P3. For all sufficiently large d, the surfaces of degree d containing a member of W form a subvariety NL(W,d) of |ØP3(d)|. We give formulas for the degree of NL(W,d), polynomial in d, when the general member C∈W is of one of the following types: a union of up to three general lines, a conic, or a twisted cubic curve.

Generic initial ideals of singular curves in graded lexicographic order

In this paper, we are interested in the generic initial ideals of singular projective curves with respect to the graded lexicographic order. Let C be a singular irreducible projective curve of degree d⩾5 with the arithmetic genus ρa(C) in Pr where r⩾3. If M(IC) is the regularity of the lexicographic generic initial ideal of IC in a polynomial ring k[x0,…,xr] then we prove that M(IC) is 1+(d−12)−ρa(C) which is obtained from the monomialxr−3xr−1(d−12)−ρa(C), provided that dimTanp(C)=2 for every singular point p∈C. This number is equal to one plus the number of secant lines through the center of general projection into P2. Our result generalizes the work of J. Ahn (2008) [1] for smooth projective curves and that of A. Conca and J. Sidman (2005) [9] for smooth complete intersection curves in P3. The case of singular curves was motivated by A. Conca and J. Sidman (2005) [9, Example 4.3]. We also provide some illuminating examples of our results via calculations done with Macaulay 2 and Singular (Decker et al., 2011 [10], Grayson and Stillman [16]).

Bases and Ideal Generators for Projective Monomial Curves

In this article we study bases for projective monomial curves and the relationship between the basis and the set of generators for the defining ideal of the curve. We understand this relationship best for curves in ℙ3 and for curves defined by an arithmetic progression. We are able to prove that the latter are set theoretic complete intersections.

Osculating spaces and diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier)

AbstractThis paper deals with some classical problems about the projective geometry of complex algebraic curves. We call locally toric a projective curve that in a neighbourhood of every point has a local analytical parametrization of type \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(t^{a_1},\dots ,t^{a_n})$\end{document}, with a1, …, an relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in P3 meets the curve only at the point of tangency. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(t^{a_1},\dots ,t^{a_n})$\end{document}, the general osculating 2‐space to a locally toric curve of genus g < 2 in P4 does not meet the curve again. The arithmetic part of the proof of this result relies on the results contained in the Appendix. By means of the same methods we give some applications and we propose possible further developments. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Intersection of ACM-curves in ℙ3

Abstract In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen–Macaulay curves in ℙ3. We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen–Macaulay curves Ct and Ct–r in ℙ3 defined by the maximal minors of a t × (t + 1), resp. (t – r ) × (t – r + 1), matrix with linear entries, provided Ct–r has no linear series of degree and dimension n ≥ t – r.

The general hyperplane section of a curve

In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of P 3 \mathbf {P}^3 . We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of P 3 \mathbf {P}^3 , arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in P n \mathbf {P}^n . For curves in P 3 \mathbf {P}^3 , we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.

Stable reflexive sheaves on smooth projective 3-folds

Motivated by Hartshorne's work on curves in P3, we study the properties of reexive rank-2 sheaves on smooth projective threefolds.