 Okay, so welcome to the speaker, Andres Rojas, who is going to talk about geometry of green semi-canonical pencils. Please. Okay, so first of all, let me thank the organizers for this school and for the opportunity to give this talk. Everything that I will explain is part of our joint work with Marty Laos and Juan Carlos Naranjo. So first of all, let me give a brief introduction to the problem. Assume that we have a double ethyl cover of a smooth curve C of genus G. Since the double cover is ethyl, the genus of CT lays 2G minus 1, then we attach to it the prime variety as the connected component of the kernel of the norm map, of the norm map, which passes through the origin of the Jacobian of C tilde, and then it comes with a principal polarization. This is the definition that we have just seen today. And then if one wants to consider the canonical model of this data divisor in the same way as in the case of a Jacobian, one considers the canonical data divisor in a torso in a translate of the original a brilliant variety, in this case of the original Jacobian, we can do the same if we want a canonical model of this, sorry, of this data divisor. Okay, is this canonical model C plus, which lives in a translate of the Jacobian of C tilde. In this canonical model, the data divisor is recovered as the intersection of the prime variety with the canonical data divisor of the curve C, but actually this intersection is not reduced, so one gets twice a principal polarization. Then map for use at this canonical model to classify the singularities of this of this data divisor into two types, stable and exception. Somehow stable correspond to line bundles with at least four global sections, these are points that have a high multiplicity at points in this in this canonical data divisor, so when we intersect we still have a singular point. And the exceptional singularity is a special condition of at points where this prime variety and this canonical data divisor somehow are tangent to itself. So we are getting a special configuration to get to get a singularity. And these exceptional singularities are described as follows. We have the pullback of a line bundle L on C without these two global sections twisted by an effective divisor on the curve C. The most elementary example of exceptional singularity is when we are pulling back a semi canonical pencil of the base curve C such that the pullback has an even number of sections. So for me as a semi canonical pencil on the curve C is a data characteristic. So it's a line bundle of degree g minus one such that it is square, it's a square is isomorphic to the canonical shift of C. And it is an even data characteristic which is effective. So it has an even number of global sections which is greater or equal to this positive. Under this condition we have an exceptional singularity. So Bobil used in the case of genus five, let me relate this construction with a work of Bobil, using the premap on on genus five or its proper version by admissible covers to prove that the Andriotti Major Lopez in A4 is the union of two reducible devices. Okay, this is the Andriotti Major Lopez in A4 which is the divisor, well it's the locus of principally polarized abelian varieties which have a singular data divisor. And Bobil showed that the two irreducible components of this Andriotti Major Lopez are J4 which are the Jacobians of curves of genus four and the divisor tetanul which parametrizes principally polarized abelian varieties whose data divisor contains not only a singular point but a point which is too torsion singular. Okay, so the key point especially for the irreducibility of tetanul was proving that the pre-image by the premap which in this genus is when one considers the proper version is subjective. This pre-image of tetanul which is the locus of curves satisfying this conditional both of a semicolonical pencil which pulls back to an even number of global sections. This is the pre-image and this pre-image is zero. So that when we apply the premap we get the divisor tetanul and it's irreducible. So we consider for an arbitrary genus gene to loci the loci we will call it TEG of a smooth covers of curves of genus G such that the base curve has a semicolonical pencil whose pullback has an even number of global sections. So this is the same condition that we were stating above just in an arbitrary genus and it's also natural to consider what happens if the number of global sections of the pullback is so. So this gives to another locus that we call TOG. Okay, so these two loci are irreducible divisors of IP and well there are closures in the delinear amount for compactification and GBAR are also well okay this appeared in a recent work with Carlos Maestro. So now the natural question is if we have these two loci we wonder about the behavior of the premap restricted to these two divisors. Okay and well this has been already studied in two particular cases as I told you Bobil studied the event case of genus five T5V to prove that the determined Andreotimia locus in E4 and also incidentally in a work of Izadi she proved that TO5 the odd case of genus of genus five dominates the whole A4 the whole modulated space A4. So in the genus five case we see that there are a big difference between these two divisors at the level of the premap because we call that Bobil show that this is mapped to TETAMUL to a divisor of A4 and on the other hand we have TO5 dominating the whole A4. So this is a very different behavior and actually the situation this is the situation in general. So for every year what we will have is that the event case the event divisor TEG is mapped to TETAMUL and this is not this is not very difficult to to prove is actually follows from the from the very definition but what happens is that TOG the odd divisors dominate AG as long as the dimension allows it. So for low dimensions between three and five TOG is dominating AG-1. Okay and actually for higher genus there are also differences but we will concentrate in these cases and specifically I will talk just to illustrate these differences about the event cases TEG in general the relation to TETAMUL and two particular cases of odd semi-clinical pensions which are in genus four and genus five somehow are the ones with which geometrically are richer. Okay so this does not consider the divisors the event divisors TEG the notation TETAMUL will stand not only for A4 but also in general it will mean the divisor of principality polarizability and varieties without the divisor that contains a two torsion single output. So then in general the pre-image of this divisor by the pre-map is essentially TEG. Okay in the cases of low genus up to genus five this pre-image is exactly TEG we call that in genus five this is Bobil. For genus data equivalency we have TEG plus some irreducible components of co-dimension three. Okay but essentially TEG is the divisorial component of the of the pre-image. Okay so let me also point out that in these irreducible components what is happening is that the green variety associated the abelian variety and not only has a two torsion singular point but also it contains a two torsion point the theta divisor with multiplicity at least three. Okay so it's a configuration that gives a more singular and theta divisor. Okay so rather than giving a proof of this which is not very difficult it's just following the classification of MAMFOR. Let me give an example of these irreducible components one of these irreducible components one g is equal to six. So for every smooth cubic three fold we can naturally associate an intermediate Jacobian with a principally polarized actually abelian variety of dimension five with a natural theta divisor a canonical theta divisor which has a unique singular point which has multiplicity three at the origin. Okay so in particular since the origin is a two torsion point of course the locus of intermediate Jacobians lies inside this device of tetanul inside a five. Okay well now we want to consider the preimage of this locus of intermediate Jacobians and compare it with the preimage of of tetanul. So if one wants to construct an intermediate Jacobian as a green variety the way is considering a line contained in the cubic three fold B. Okay we consider a supplementary plane P2 in the ambient P4 so that this plane is parametrizing all the planes containing the line L and what we what we are doing is projection from B into this plane so simply what we are doing is for every point of B we consider the plane which is obtained from joining the point with the line okay this is of course rational it's not defined along L that one can blow up and this is a morphine this often determinacy locus. Okay well essentially what we are doing what are the fibers of this map to every plane containing L we intersect in the cubic three fold with this plane and we are considering the intersection the intersection is the line L and the residual okay well then the locus of planes through L is parametrized by this P2 such that this residual conic is reducible so that it breaks up it breaks into two lines is a plane quintic associated to this conic vandal structure. So essentially to every point of this conic that we are doing is considering the plane obtained from joining with L in this plane intersects the cubic three fold in L and two further lines okay well if we didn't know this plane quintic by QL there is a natural covering QL tilde let's call it that essentially puts over over every point of QL the two lines the two residual lines are rising in the intersection and the prime variety of this cover is the intermediate Jacobian and this is all the possible ways one can construct the intermediate Jacobian as a as a prime variety. So if one considers the pre-match of all intermediate Jacobians one has the locus formed by covers of plane quintics satisfy an aparatical mixture okay I mean not every cover of a plane quintic arises in this way but if one puts a parity condition here then one recovers these constructions we call it RQ RQ minus following the don't have any notation I think well then what happens is that this loci RQ minus is an irreducible component of the pre-match of the TANU because RQ minus is not contained in TE6 okay mainly because the the general quintic has no semi-canal complex so this is an irreducible component and has dimension 12 because the locus of quintics well the model space of plane quintics has dimension 12 and this has a indeed co-dimension three in R6 and we have a component whose data divisor has a point has a point with higher multiplicity so this is all I wanted to say about the even case let's move to the odd case TO4 and as I told you before this map is going to be dominant actually okay in contradiction in contradiction to the to the even case the result is that this premap restricted to TO4 is subjective and actually the fiber the general fiber of this map fiber at the general Jacobian is the complement in the projective plane of the union of the canonical model of X which is a quartic plane curve and the 28th v tangent lines to this plane quartic okay essentially the idea to prove this is to use Recija's trigonal construction this is an instance of the polygonal constructions that Irene already introduced this morning the philosophy of the Recija's trigonal construction is that if we have a double cover of a trigonal curve then it's pretty bad it's pretty variety it's going to be the Jacobian of a tetragonal curve and conversely so the idea is that we have a double etal cover of a trigonal curve this is three to one this is two to one then one can associate at the trigonal curve X such that the Jacobian of X is is going to be the premarital discord and under certain generality conditions on the curve X one can reverse this construction okay so let's prove it in a in a more in a more sophisticated way but the diagram is going to be the same if we consider the modular space of objects like the ones I consider here in the left so we have a cover of a smooth curve of genus four that I will assume assume also that this is not hyper elliptic and though we with a trigonal line bundle on the curve we consider this modular space and on the left we consider and the right sorry we consider the modular space of objects like the ones presented here so we have a pair format by a curve of genus three and a g14 on X and I will assume it's base point three in order to give a morphism to p1 okay but the u14 must not not this doesn't need to be complete so then what the trigonal construction says is that we start with the modular space parameterizing the objects in the left if we forget the trigonal curve and we apply the premar to the cover then we are going to get the same ambient variety as if we apply the trigonal construction forget the trigonal line bundle and then we simply consider it okay well then the idea to find the the fiber restricted to t04 at a general curve X by this diagram that actually is a nice morphism if we restrict to a certain open subset here okay then we have to leave this element this curve X to this open subset come back the arzillas forget the trigonal line bundle and once we are here intersect with t04 well the method to do this is that we can consider true involution sigma on both modular spaces I will not give the the test because I don't want to be over time but essentially the idea is that they commute with with rathias trigonal construction so the key point is that when we are doing all this way around in order to arrive here and intersect with t04 once we are at this point of the modular space the pre-image of this prediction file of t04 is one of the fixed set of side that this involution sigma okay so since these involutions are compatible it must come back it must come from sorry from a fixed loci by tau okay and once we are in tau this is related to the geometry of x and here it's much easier to consider this fixed loci okay so this fixed component corresponds to the to the g1 force which are not complete on x so these are these are parameterized by the project plane because the g1 force non-complete are continuing the canonical the canonical bundle and then we have to discard the points of x in this p2 because they are giving base point base points in our g14 and we have to discard also the 20 determinant lines because are because are lying outside these open stops okay so i won't write anything else about this i'm sorry for the explanation i can give you more details if you want but this is more or less eight so now let me move to the case of genius five which is probably the most interesting one we want to understand how is the premap the mobile's proper version if you want of the premap restricted to t05 okay first of all let me say that donating in the description of these of these premaps found our our rational map between a4 and the modulus space of pairs format by a smooth cubic three four b and a two torsion point in its intermediate jacobia which has a certain parity condition okay and this is actually by russia is adding gave an explicit description of this by russia map and found the indeterminacy locals and somehow incidentally in this study one can find the following result of his study is that this premap restricted to t05 is rejected okay now actually the fiber at the general element of this rc plus is a partial desingularization of the curve of a curve in the final surface of lines of the cubic truth okay so this is the final surface and the curve that we are considering parametrized lines such that when we are considering all the planes through l there exists one such that the residual intersection is a double line is a double line r okay so with regard to the to the previous construction of the conic bundle this means that the discriminant quintic has a point where actually there is only one point lying over it there is a double line two lines are somehow collapsing and this produces a note on the on the corresponding quintic but the cover is admissible anyway okay the idea of the proof goes as follows don't have you describe the fiber of the premap in general not restricted to a divisor at the general element of this modular space by rational 284 as a double cover of the final surface of lines of b this is what i denote by f the fb tilde it is the 2 to 1 et al cover which actually is defined by these two torsion points so this information that gives the two torsion points but one has the curved gamma inside the final surface of lines and one can see that the pre-image of this curve gamma by this double et al cover has three reusable components one of one of them is the intersection of t05 with this general fiber so actually this component is the general fiber restricted to t05 of course and another component which is the intersection of delta zero ram this is the divisor of bobbin admissible covers of nodal curves so here are curves of singular of covers sorry of singular curves and this is the intersection of this divisor with the general actually the nagi's description of the involution here is changing sheets for the cover which also was described alternatively by zadi both descriptions are very geometric and give that these two components are exchanged by hand okay so in particular it it follows that gamma must be same because the final surface of a general cubic three four has picard run one so then what you get is that the curve gamma example the final surface and if it's pretty much where where smooth then it should be reduced okay these are consequence of of hoaching next year so gamma must be single at the picture you have to to regarded as follows we have the curve gamma and here we have the two components some of the notes of sorry some of some of the singular points of gamma leave two notes of these components but there are also singular points of gamma which actually leave two intersection points of the components okay this is the reason why this general fiber one of the two components is a partial desingualization of the curve gamma okay this is what sorry sorry andres you have three minutes left so i will finish thanks so just the final part is we cannot take more information about gamma with analysis of the divisors with the divisor 205 okay so we have described gamma this is the definition there is a plane with a residual conic which is a double line but in the same way one can define a natural counterpart which is a curve gamma plane okay it's formed by lines are such that there exists a two plane which intersects and the cubic three four in two times this line okay this somehow a natural a natural counterpart this is called classically the lines of second okay this is classically studied it's called gamma prima is known to be smooth for a general cubic three four b and also it's it's it's numerical class also one can it's easy to prove that if since a cubic three four b is smooth then one have one has a natural morphism from gamma prima to alpha okay because for every line are one can only have a plane satisfying that the intersection along this plane is is double for the line one cannot have two planes so the situation is that the curve gamma prima is smooth and actually one can prove that this is the normalization of gamma which i told you that is always singular and with a bit more of work one can prove that the only notes of that the only singularities of gamma are notes okay and these notes have a geometric meaning because these are the points which are which have two parameters in gamma prima by this by this by this normalization so i will not make the picture but you can imagine the situation of a line and two distinct planes with residual intersections double lines okay so once we are here a natural numerative question is what is the number of notes of gamma for a general curve so what is the exact number of lines satisfying this configuration and one can prove that gamma is numerically equivalent to eight times the canonical divisor and we will deduce from it the number of notes of gamma okay i won't give the details of the proof but only let me let me tell you that the way of finding this numerical class is somehow surprising at least for us because what we used is that donagis description of the of the general fiber realizes this double cover of the final surface of lines in this moderate space of admissible cover covers or if you want in the in the delinium for compactification now the fact that this space has a well has a well known picard group description the canonical class is also well known allows us to find properties of the well of the double cover but also the final surface of lines which somehow if one tries to see directly in the cubics refolders cannot cannot detect okay so expressing this funnel as a as a general fiber allows to give more information also about the cubics it's not only to say things about the print out okay and here one uses to prove this that the homology class of t o five is well known a one can compare it with with the canonical class okay using that we know the numerical class we can compute its arithmetic genus the geometric genus of gamma is that of gamma prime because these curves are are birational and computing this difference is numerical class is quite high in comparison with that of gamma prime one obtains that the exact number of nodes is 1000 485 okay so if you want to try translate these nodes into lines there are there is this number of lines for a general cubics refold satisfying this this condition and well this is all I wanted to say so thank you for for staying here thank you very much address for this very interesting talk so is this there any question I have lots of questions but maybe if you have some time tomorrow I would like to understand better your beautiful construction with the based on the trigonal reseal of trigonal construction so maybe if tomorrow you can explain it to me I will be very happy I will be glad to explain thanks okay so any remark question okay so if not we are all a little bit tired but that was a very nice talk both of the talks were very nice yes my compliments to everybody today talks were very beautiful yes