arXiv subjects
Publications and source records attributed to Enrico Schlesinger.
We introduce a notion of good cohomology for multiple lines in $\mathbb{P}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a surface of degree less than d, and of maximal arithmetic genus is not irreducible already for d=4 and d=5.
Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when d=s or d is at least 2s-1. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.
Let $S$ be a smooth algebraic surface in $\mathbb{P}^3(\mathbb{C})$. A curve $C$ in $S$ has a cohomology class $η_C \in H^1 \hspace{-3pt}\left( Ω^1_S \right)$. Define $α(C)$ to be the equivalence class of $η_C$ in the quotient of $H^1 \hspace{-3pt}\left( Ω^1_S \right)$ modulo the subspace generated by the class $η_H$ of a plane section of $S$. In the paper "Reconstructing subvarieties from their periods" the authors Movasati and Sertöz pose several interesting questions about the reconstruction of $C$ from the annihilator $I_{α(C)}$ of $α(C)$ in the polynomial ring $R=H^0_*(\mathcal{O}_{\mathbb{P}^3})$. It contains the homogeneous ideal of $C$, but is much larger as $R/I_{α(C)}$ is artinian. We give sharp numerical conditions that guarantee $C$ is reconstructed by forms of low degree in $I_{α(C)}$. We also show it is not always the case that the class $α(C)$ is \textit{perfect}, that is, that $I_{α(C)}$ could be bigger than the sum of the Jacobian ideal of $S$ and of the homogeneous ideals of curves $D$ in $S$ for which $I_{α(D)}=I_{α(C)}$.
Let $P_{\text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $\mathbb{P}^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{\text{MAX}}(d,s)$ has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family $\mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $\mathcal{C}$ has good cohomological properties. With the aid of \emph{Macaulay2} we checked the validity of the conjecture for $s \leq 100$. From the conjecture it would follow that $P(d,s)= P_{\text{MAX}}(d,s)$ for $d=s$ and for every $d \geq 2s-1$.
Let $H_{d,g}$ denote the Hilbert scheme of locally Cohen-Macaulay curves of degree $d$ and genus $g$ in projective three space. We show that, given a smooth irreducible curve $C$ of degree $d$ and genus $g$, there is a rational curve $\{[C_t]: t \in \mathbb{A}^1\}$ in $H_{d,g}$ such that $C_t$ for $t \neq 0$ is projectively equivalent to $C$, while the special fibre $C_0$ is an extremal curve. It follows that smooth curves lie in a unique connected component of $H_{d,g}$. We also determine necessary and sufficient conditions for a locally Cohen-Macaulay curve to admit such a specialization to an extremal curve.
Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the connectedness of the Hilbert scheme of curves in projective 3 space" Comm. Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM workshop Components of Hilbert Schemes available at http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of curves whose general member is a union of d disjoint lines on a smooth quadric surface and whose special member is a locally Cohen-Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d, we construct a family with the required properties, whose special fiber is an extremal curve in the sense of Martin-Deschamps and Perrin. From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.
We give a new example of a curve C algebraically, but not rationally, uniformized by radicals. This means that C has no map onto the projective line P^1 with solvable Galois group, while there exists a curve C' that maps onto C and has a finite morphism to P^1 with solvable Galois group. We construct such a curve C of genus 9 in the second symmetric product of a general curve of genus 2. It is also an example of a genus 9 curve that does not satisfy condition S(4,2,9) of Abramovich and Harris.
Let C be an ACM (projectively normal) nondegenerate smooth curve in projective 3-space, and suppose C is general in its Hilbert scheme - this is irreducible once the postulation is fixed. Answering a question posed by Peskine, we show the gonality of C is d-l, where d is the degree of the curve, and l is the maximum order of a multisecant line of C. Furthermore l=4 except for two series of cases, in which the postulation of C forces every surface of minimum degree containing C to contain a line as well. We compute the value of l in terms of the postulation of C in these exceptional cases. We also show the Clifford index of C is equal to the gonality minus 2.
Zariski proved the general complex projective curve of genus g>6 is not rationally uniformized by radicals, that is, admits no map to the projective line whose Galois group is solvable. We give an example of a genus 7 complex projective curve Z that is not rationally uniformized by radicals, but such that there is a finite covering Z' -> Z with Z' rationally uniformized by radicals. The curve providing the example appears in a paper by Debarre and Fahlaoui where a construction is given to show the Brill Noether loci W_d(C) in the Jacobian of a curve C may contain translates of abelian subvarieties not arising from maps from C to other curves.
The uniform position principle states that, given an irreducible nondegenerate curve C in the projective r-space $P^r$, a general (r-2)-plane L is uniform, that is, projection from L induces a rational map from C to $P^1$ whose monodromy group is the full symmetric group. In this paper we show the locus of non-uniform (r-2)-planes has codimension at least two in the Grassmannian for a curve C with arbitrary singularities. This result is optimal in $P^2$. For a smooth curve C in $P^3$ that is not a rational curve of degree three, four or six, we show any irreducible surface of non-uniform lines is a Schubert cycle of lines through a point $x$, such that projection from $x$ is not a birational map of $C$ onto its image.
In this paper we determine the irreducible components of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g. We show that these Hilbert schemes are connected, in spite of having about g^2/24 irreducible components. For g < -2 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Ait-Amrane and Perrin.
Let F be a smooth surface in a smooth projective threefold T, and let X=2F be the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve C in X gives rise to two effective divisors on F, namely the curve part P of the intersection of C and F, and the curve R residual in C to this intersection. We show that a general deformation of R on F lifts to a deformation of C on X when a certain cohomology group vanishes. In our paper "Hilbert Schemes of Degree Four Curves" we use this result to prove the connectedness of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g.
We study locally Cohen-Macaulay curves in projective three-space which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes of locally Cohen-Macaulay curves in 2H of given degree and arithmetic genus. We show that these Hilbert schemes are connected. We also discuss the Rao modules of these curves, and liaison and biliaison equivalence classes.