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Let $X$ be a double EPW sextic, and $\iota$ its anti-symplectic involution. We relate the $\iota$-anti-invariant part of the Chow group of zero-cycles of $X$ with Voisin's rational orbit filtration. For a general double EPW sextic $X$, we also relate the anti-invariant part of the Chow motive of $X$ with the motive of a Gushel-Mukai fourfold. As an application, we obtain a similar result for certain Fano varieties of lines in cubics with infinite-order birational automorphisms., Comment: 21 pages, comments very welcome ! arXiv admin note: substantial text overlap with arXiv:2307.15240

In this paper we explore the intersection of the Hassett divisor $\mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $\mathcal C_i$. Notably we study the irreducible components of the intersections with $\mathcal{C}_{12}$ and $\mathcal{C}_{20}$. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of $P$ with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off $P$, or by finding examples of reducible one-apparent-double-point surfaces inside $X$. Finally, via some Macaulay computations, we give explicit equations for cubics in each component., Comment: arXiv admin note: text overlap with arXiv:1504.05863

Let S be a K3 surface obtained as triple cover of a quadric branched along a genus 4 curve. Using the relation with cubic fourfolds, we show that S has finite dimensional motive, in the sense of Kimura. We also establish the Kuga-Satake Hodge conjecture for S, as well as Voisin'conjecture concerning zero-cycles. As a consequence, we obtain Kimura finite dimensionality, the Kuga-Sataka Hodge conjecture, and Voisin's conjecture for 2 (9-dimensional) irreducible components of the moduli space of K3 surfaces with an order 3 non-symplectic automorphism., Comment: 20 pages, to appear on Rend. Circ. Mat. di Palermo

Given a smooth genus two curve $C$, the moduli space SU$_C(3)$ of rank three semi-stable vector bundles on $C$ with trivial determinant is a double cover in $\mathbb{P}^8$ branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of $C$. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of $GL_9$, initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}. Starting from a general trivector $v$ in $\wedge^3\mathbb{C}^9$, we construct a Fano manifold $D_{Z_{10}}(v)$ in $G(3,9)$ as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SU$_C(3)$. We deduce that $D_{Z_{10}}(v)$ is isomorphic to the odd moduli space SU$_C(3, \mathcal{O}_C(c))$ of rank three stable vector bundles on $C$ with fixed effective determinant of degree one. We deduce that the intersection of $D_{Z_{10}}(v)$ with a general translate of $G(3,7)$ in $G(3,9)$ is a K3 surface of genus $19$.

Let $X^{1,n}_r$ be the blow-up of $\mathbb{P}^1\times\mathbb{P}^n$ in $r$ general points. We describe the Mori cone of $X^{1,n}_r$ for $r\leq n+2$ and for $r = n+3$ when $n\leq 4$. Furthermore, we prove that $X^{1,n}_{n+1}$ is log Fano and give an explicit presentation for its Cox ring., Comment: 12 pages

Given a smooth genus three curve $C$, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in $\mathbb{P}^8$ as a hypersurface whose singular locus is the Kummer threefold of $C$; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU$_C(2, L)$, the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2, 8)$. In fact, each point $p \in C$ defines a natural embedding of SU$_C(2, \mathcal{O}(p))$ in $G(2, 8)$. We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of SU$_C(2, \mathcal{O}(p))$, and thus deserves to be coined the Coble quadric of the pointed curve $(C, p)$.

We show that a wide range of Fano varieties of K3 type, recently constructed by Bernardara, Fatighenti, Manivel and Tanturri, have a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. It follows that the Chow ring of these Fano varieties behaves like that of K3 surfaces. As a side result, we obtain some criteria for the Franchetta property of blown-up projective varieties., Comment: 19 pages, comments welcome. Version 2: revised according to referee's comments. To appear in "Perspectives on four decades: Algebraic geometry 1980---2020. In memory of Alberto Collino'', Trends in Mathematics, Birkha\"user

Gushel-Mukai sixfolds are an important class of so-called Fano-K3 varieties. In this paper we show that they admit a multiplicative Chow-K\"unneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double EPW sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of Gushel-Mukai sixfolds., Comment: 22 pages, all comments welcome. Version 2: revised according to referees' comments, to appear in Math. Nachrichten

Let $\mathcal{C}_d$ denote Hassett's Noether-Lefschetz divisor in the moduli space of cubic fourfolds given by cubics with intersection form of discriminant $d$. We exhibit arithmetic conditions on 20 indexes $d_1,\dots, d_{20}$ that ensure that the divisors $\mathcal{C}_{d_1},\dots,\mathcal{C}_{d_{20}}$ all intersect one another. This allows us to produce examples of rational cubic fourfolds with an associated K3 surface with any prescribed N\'eron-Severi rank. Then we apply this result to show that there exist one dimensional families of cubic fourfolds with finite dimensional Chow motive of Abelian type inside every divisor of special cubic fourfolds. This also implies Abelianity and finite dimensionality of the motive of related Hyperk\"ahler varieties, such as the Fano variety of lines and the LLSvS 8fold. A similar remark, plus some elementary Brill-Noether geometry of the associated K3 surfaces, allows us to show the Abelianity of the motive of an infinity of LSV 10folds, associated to Pfaffian cubic fourfolds. After that, starting from known 4-dimensional families of K3 surfaces, we construct two families $\mathcal{F}$ and $\mathcal{G}$ of cubic fourfolds whose motive is of Abelian type. Cubics from the first family are smooth, and their Chow motive is finite dimensional and Abelian. Those from the second family are singular, and their motives are Schur-finite and Abelian in Voevodsky's triangulated category of motives., Comment: 30 pages, it merges arXiv:2007.07193, arXiv:2004.08446 plus several new results

The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently., Comment: 14 pages, improved results

In this short note, we show that there exist one dimensional families of cubic fourfolds with Chow motive of abelian type and finite dimensional inside every Hassett divisor of special cubic fourfolds. This also implies abelianity and finite dimensionality of the motive of related Hyperk\"ahler varieties, such as the Fano variety of lines and the LLSvS 8fold., Comment: 9 pages

We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott K\"ahler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson-Schechtman and Bloch-Esnault., Comment: 41 pages, final version; to appear in J. Inst. Math. Jussieu

The aim of this short note is to define the \it universal cubic fourfold \rm over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett divisors $\mathcal{C}_d$, in the range $8\leq d \leq 42$. By applying inductively this argument, we are able to show that, in the same range of values, $\mathcal{C}_{d,n}$ is unirational for all integer values of $n$. Finally, we observe that for explicit infinitely many values of $d$, the universal cubic fourfold over $\mathcal{C}_d$ can not be unirational., Comment: 14 pages

Let $C$ be a hyperelliptic curve of genus $g\geq 3$. In this paper we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on $C$ with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//PGL(2)$. Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree two osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$-folds over $\mathbb{P}^g$ inside the ramification locus of the theta map., Comment: 22 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1802.00654

In this note we introduce the transcendental part $t(X)$ of the motive of a cubic fourfold $X$ and prove that it is isomorphic to the (twisted) transcendental part $h_2^{tr}(F(X))$ in a suitable Chow-K\"unneth decomposition for the motive of the Fano variety of lines $F(X)$. Then we prove that $t(X)$ is isomorphic to the Prym motive associated to the surface $S_l \subset F(X)$ of lines meeting a general line $l$. If $X$ is a special cubic fourfold in the sense of Hodge theory, and $F(X)\cong S^{[2]}$, with $S$ a $K3$, then we show that $t(X) \cong t_2(S)(1)$, where $t_2(S)$ is the transcendental motive. Therefore the motive $h(X)$ is finite dimensional if and only if $S$ has a finite dimensional motive. If $X$ is very general then $t(X)$ cannot be isomorphic to the (twisted) transcendental motive of a surface. We relate the existence of an isomorphism $t(X) \cong t_2(S)(1)$ to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we consider the case of cubic fourfolds X admitting a fibration over $\mathbf{P}^2$, whose fibers are either quadrics or del Pezzo surfaces of degree 6, and prove the isomorphism $t_2(S)(1) \cong t(X)$, with $S$ a K3 surface., Comment: 27 pages

We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle., Comment: 14 pages

In this paper we study the geometry of GIT configurations of $n$ ordered points on $\mathbb{P}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $(\mathbb{P}^1)^n//PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of automorphisms., Comment: 22 pages (fixed one small inaccuracy)

In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus is not open in $\mathcal C_{14}$., Comment: 24 pages. Final, rewritten and expanded version with some new results, to appear on Math. Annalen

In this paper we prove Homological Projective Duality for crepant categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a n x m matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality and categorical representability in codimension two for determinantal varieties., Comment: 23 pages. To appear on Advances in Math

We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a $(d_1,d_2)$-polarized abelian surface, we show how the parities of the $d_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d_1$ and $d_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geometry and the Kodaira dimension of these moduli spaces., Comment: Final version. To appear in Commentarii Mathematici Helvetici (CMH)

In this paper we study the topology of the stack $\mathcal{T}_g$ of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes $\mathcal{T}_g$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $\mathcal{T}_g$, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group $\mathcal{M}ap_g$ of Riemann surfaces of genus g., Comment: To appear on Selecta Math

We describe a relation between the invariants of $n$ ordered points in $P^d$ and of points contained in a union of linear subspaces $P^{d1}\cup P^{d2} \subset P^d$. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational normal curve degenerates, then the admissible covers limit is a cyclic cover with weights as in this attaching map. We find that both GIT polarizations and the Hodge class for families of cyclic covers yield line bundles on $\bar{M}_{0,n}$ with functorial restriction to the boundary. We introduce a notion of divisorial factorization, abstracting an axiom from rational conformal field theory, to encode this property and show that it determines the isomorphism class of these line bundles. As an application, we obtain a unified, geometric proof of two recent results on conformal block bundles, one by Fedorchuk and one by Gibney and the second author., Comment: 17 pages, 3 figures

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve of the quadric bundle, we prove that the general member is both pfaffian and has nontrivial Brauer class. Such cubic fourfolds also provide twisted derived equivalences between K3 surfaces of degree 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds., Comment: 13 pages, minor revision, final version

Let X -> Y be a fibration whose fibers are complete intersections of two quadrics. We develop new categorical and algebraic tools---a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting---to study semiorthogonal decompositions of the bounded derived category of X. Together with new results in the theory of quadratic forms, we apply these tools in the case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if Y is the projective line over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X., Comment: 43 pages, changes made and some material added and corrected in sections 1, 4, and 5; this is the final version accepted for publication at Journal de Math\'ematiques Pures et Appliqu\'ees

Let $C$ be an algebraic smooth complex curve of genus $g>1$. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on $C$ and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients $(\PP^{r-1})^{rg}//PGL(r)$, where $r$ is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, $\alpha$-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when $C$ has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces., Comment: Supersedes 1002.4382

We define, basing upon semiorthogonal decompositions of $\Db(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions.

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal., Comment: New version; now also the case of cubic degeneration in P^2 is described in detail. 23 Pages

Let $C$ be an algebraic smooth complex genus $g>1$ curve. The object of this paper is the study of the birational structure of the coarse moduli space $U_C(r,0)$ of semi-stable rank r vector bundles on $C$ with degree 0 determinant and of its moduli subspace $SU_C(r)$ given by the vector bundles with trivial determinant. Notably we prove that $U_C(r,0)$ (resp. $SU_C(r)$) is birational to a fibration over the symmetric product $C^(rg)$ (resp. over $P^{(r-1)g}$) whose fibres are GIT quotients $(P^{r-1})^{rg}//PGL(r)$. In the cases of low rank and genus our construction produces families of classical modular varieties contained in the Coble hypersurfaces., Comment: 16 pages, corrected references

Let $\cM_{0,n}$ the moduli space of $n$-pointed rational curves. The aim of this note is to give a new, geometric construction of $\cM_{0,2n}^{GIT}$, the GIT compacification of $\cM_{0,2n}$, in terms of linear systems on $\PP^{2n-2}$ that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps between $\bar{\cM}_{0,2n+1}$ and $\bar{\cM}_{0,2n}$. The construction is performed via a study of the so-called $\textit{contraction}$ maps from the Knudsen-Mumford compactification $\bar{\cM}_{0,2n}$ to $\cM_{0,2n}^{GIT}$ and of the canonical forgetful maps. As a side result we also find a linear system on $\bar{\cM}_{0,2n}$ whose associated map is the contraction map $c_{2n}$., Comment: New version: corrected typos, added contextualization and relations with previous GIT constructions and with linear systems on the Knudsen compactification

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors., Comment: 25 pages, new version. Added a couple of new Prop.s and lots of references, corrected inaccuracies and typos

In this paper we give a presentation of the stack of trigonal curves as a quotient stack, and we compute its Picard group., Comment: Corrected typos, added references and improved the exposition, following the suggestions of the referees. To appear in Transactions of the American Mathematical Society"

This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call \textit{locally toric} a projective curve that in a neighbourhood of every point has a local analytical parametrization of type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in $\bP^3$ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \cite{kaji} by H.Kaji where the same result is proven for any curve in $\bP^3$ such that every branch is smooth. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations $(t^{a_1},...,t^{a_n})$, the general osculating 2-space to a locally toric curve of genus $g<2$ in $\bP^4$ does not meet the curve again. The arithmetic part of the proof of this result relies on the Appendix \cite{cz:rk} to this paper. By means of the same methods we give some applications and we propose possible further developments., Comment: 23 pages

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $\omega$ of $C$ with $\omega_C^{-1}$. In this paper we study the classifying rational map $\phi: \pr Ext^1(\omega,\omega^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface $S$ and $\su$ at the point $p$ corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tilde{\phi}: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components., Comment: 29 pages

The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A\_2(3)^- parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A\_2(3)^- is rational. Then, given a genus 2 curve C, we give an interpretation of the Weddle surface as a moduli space of extensions classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf \omega of C with \omega^{-1}. This in turn allows to see the Weddle surface as a hyperplane section of the secant variety Sec(C) of the curve C tricanonically embedded in P^4., Comment: 35 pages, to appear in Advances in Geometry, part of the author's Phd Thesis