Severi Varieties Research Articles

Absolutely split locally free sheaves on proper k-schemes and Brauer–Severi varieties

We classify absolutely split vector bundles on proper k-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer–Severi varieties.

Ulrich bundles on Brauer–Severi varieties

We prove the existence of Ulrich bundles on any Brauer–Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer–Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer–Severi variety X X , at least if d i m ( X ) = p − 1 \mathrm {dim}(X)=p-1 for an odd prime p p . This formula implies that the rank of any Ulrich bundle on such a Brauer–Severi variety X X must be a multiple of the period.

A uniform characterisation of the varieties of the second row of the Freudenthal–Tits magic square over arbitrary fields

We characterise the projective varieties related to the second row of the Freudenthal–Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the 27 -dimensional \mathrm{E_6} module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.

Nonemptiness of severi varieties on enriques surfaces

Abstract Let $(S,L)$ be a general polarised Enriques surface, with L not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible nodal curves in the linear system $|L|$ , that is, for any number of nodes $\delta =0, \ldots , p_a(L)-1$ . This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande–Schmitt, under the additional condition of non-2-divisibility.

Severi varieties on blow‐ups of the symmetric square of an elliptic curve

AbstractWe prove that certain Severi varieties of nodal curves of positive genus on general blow‐ups of the twofold symmetric product of a general elliptic curve are nonempty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces.

On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.

Nodal elliptic curves on K3 surfaces

Let (X, L) be a general primitively polarized K3 surface with \(c_1(L)^2 = 2g-2\) for some integer \(g \ge 2\). The Severi variety \(V^{L,\delta } \subset |L|\) is defined to be the locus of reduced and irreducible curves in \(|L|\) with exactly \(\delta \) nodes and no other singularities. When \(\delta =g\), any curve \(C \in V^{L,g}\) is a rational curve; in fact, Chen (Math Ann 324:71–104, 2002) has shown that all rational curves in \(|L|\) are nodal, and the number of such rational curves is given by the Yau-Zaslow formula [20]. In this paper, we consider the next case where \(\delta = g-1\) and the Severi variety \(V^{L,g-1}\) parametrizing nodal elliptic curves is of dimension 1. Let \(\overline{V}^{L,g-1} \subset |L|\) denote the Zariski closure. For a reduced curve C, we define the geometric genus of C to be the sum of the genera of the irreducible components of the normalization. We prove that the geometric genus of the closure \(\overline{V}^{L,g-1} \subset |L|\) is bounded from below by \(O(e^{C\sqrt{g}})\).

On Severi varieties as intersections of a minimum number of quadrics

Let \({\mathscr{V}}\) be a variety related to the second row of the Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely in \({\mathscr{V}}\) if and only if there exists a subspace of projective dimension \(N-M\) in the secant variety disjoint from the Severi variety. We present some examples of such subspaces of relatively large dimension. In particular, over the real numbers we show that the Cartan variety (related to the exceptional group \({E_6}\)\((\mathbb R)\)) is the set-theoretic intersection of 15 quadrics.

Genus two curves on Abelian surfaces

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves are nodal. We prove that this holds true if and only if d_2 is not divisible by 4. In the cases where d_2 is a multiple of 4, we exhibit genus 2 curves in |L| that have a triple, 4-tuple or 6-tuple point. We show that these are the only possible types of unnodal singularities of a genus 2 curve in |L|. Furthermore, with no assumption on d_1 and d_2, we prove the existence of at least a nodal curve in |L|. As a corollary, we obtain nonemptiness of all Severi varieties on general abelian surfaces and hence generalize [KLM, Thm 1.1] to nonprimitive polarizations.

Degeneration of curves on some polarized toric surfaces

Abstract We address the following question: Given a polarized toric surface ( S , L ) {(S,{\mathcal{L}})} , and a general integral curve C of geometric genus g in the linear system | L | {|{\mathcal{L}}|} , do there exist degenerations of C in | L | {|{\mathcal{L}}|} to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show the irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.

A note on the Severi problem for toric surfaces

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

Severi dimensions for unicuspidal curves

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. cusps; and to do so, we stratify cusps according to value semigroup. We show that generalized Severi varieties of maps P1→Pn with images of fixed degree and arithmetic genus are often reducible whenever n≥3. We also prove that the Severi variety of degree-d maps with a hyperelliptic cusp of delta-invariant g≪d is of codimension at least (n−1)g inside the space of degree-d holomorphic maps P1→Pn; and that for small g, the bound is exact, and the corresponding space of maps is the disjoint union of unirational strata. Finally, we conjecture a generalization for unicuspidal rational curves associated to an arbitrary value semigroup.

The Severi problem for abelian surfaces in the primitive case

Severi varieties are roughly moduli spaces of curves of a fixed homology class and geometric genus on a projective surface. In this paper, we determine the irreducible components of the Severi varieties of primitive (indivisible) homology class curves on a general abelian surface, answering a question of Knutsen, Lelli-Chiesa, and Mongardi. More specifically, we prove that the irreducible components are completely determined by the maximal factorization of the morphisms from the normalized curves into the abelian surface.

On the section conjecture and Brauer\u2013Severi varieties

J. Stix proved that a curve of positive genus over mathbb {Q} which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction R_{k/mathbb {Q}}X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction {text {cor}}_{k/mathbb {Q}}([P])in {text {Br}}(mathbb {Q}) is non-trivial, then X satisfies the section conjecture.

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer–Severi varieties

We show that the derived category of Brauer–Severi curves satisfies the Jordan–Holder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field k. Moreover, we show that a n-dimensional Brauer–Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length n+1, at least in characteristic zero. We conjecture that Brauer–Severi varieties X satisfy rdimcat(X)= ind(X)−1, provided period equals index, and prove this in the case of curves, surfaces and for Brauer–Severi varieties of index at most three. We believe that the results for curves are known to the experts. We nevertheless give the proofs, adding to the literature.

A Characterization of Secant Varieties of Severi Varieties Among Cubic Hypersurfaces

Abstract It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

On the birational geometry of Hilbert schemes of points and Severi divisors

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of and when there is a birational morphism between surfaces. In this setting, embeds in and we ask if the restriction of the stable base locus decomposition of yields the respective decomposition in if the weak Lefschetz Principle holds. Even though the stable base loci in fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of to is equal to the stable base locus decomposition of We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that X is the projective plane, these divisors yield walls of the SBLD for some cases.

ON THE POSITION OF NODES OF PLANE CURVES

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^{2}$ of degree $n$. This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$, we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in $\text{Pic}(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz–Harris [‘Geometry of the Severi variety’, Trans. Amer. Math. Soc.309 (1988), 1–34] about whether the canonical class of the Severi variety is effective.

Cubic forms having matrix factorizations by Hessian matrices

Using a part of XJC-correspondence by Pirio and Russo, we classify cubic forms f f whose Hessian matrices induce matrix factorizations of themselves. When it defines a reduced hypersurface, it satisfies the “secant–singularity” correspondence, that is, it coincides with the secant locus of its singular locus. In particular, when f f is irreducible, its singular locus is one of the four Severi varieties.

Comment on: On the irreducibility of the Severi variety of nodal curves in a smooth surface, by E. Ballico

In this short note, I point out that results of Ballico and Kool–Shende–Thomas together imply that on K3, Enriques, and Abelian surfaces, if L is a very ample and $$(2p_a(L)-2g-1)$$-spanned line bundle, then the equigeneric Severi variety $$V_{g}(L)$$ of all curves in |L| having genus g is non-empty, irreducible, of the expected dimension, and its general member is a $$(p_a(L)-g)$$-nodal curve.